diff --git a/Mathbin/Algebra/Category/Module/Kernels.lean b/Mathbin/Algebra/Category/Module/Kernels.lean index a60395901d..9aad56d838 100644 --- a/Mathbin/Algebra/Category/Module/Kernels.lean +++ b/Mathbin/Algebra/Category/Module/Kernels.lean @@ -30,11 +30,23 @@ section variable {M N : ModuleCat.{v} R} (f : M ⟶ N) +/- warning: Module.kernel_cone -> ModuleCat.kernelCone is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.KernelFork.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) M N f +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.KernelFork.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u2, u1} R _inst_1)) M N f +Case conversion may be inaccurate. Consider using '#align Module.kernel_cone ModuleCat.kernelConeₓ'. -/ /-- The kernel cone induced by the concrete kernel. -/ def kernelCone : KernelFork f := KernelFork.ofι (asHom f.ker.Subtype) <| by tidy #align Module.kernel_cone ModuleCat.kernelCone +/- warning: Module.kernel_is_limit -> ModuleCat.kernelIsLimit is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.IsLimit.{0, u2, 0, max u1 (succ u2)} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Limits.parallelPair.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) M N f (OfNat.ofNat.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (OfNat.mk.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (Zero.zero.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) M N))))) (ModuleCat.kernelCone.{u1, u2} R _inst_1 M N f) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.IsLimit.{0, u2, 0, max u1 (succ u2)} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Limits.parallelPair.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) M N f (OfNat.ofNat.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (Zero.toOfNat0.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u2, u1} R _inst_1)) M N)))) (ModuleCat.kernelCone.{u1, u2} R _inst_1 M N f) +Case conversion may be inaccurate. Consider using '#align Module.kernel_is_limit ModuleCat.kernelIsLimitₓ'. -/ /-- The kernel of a linear map is a kernel in the categorical sense. -/ def kernelIsLimit : IsLimit (kernelCone f) := Fork.IsLimit.mk _ @@ -49,11 +61,23 @@ def kernelIsLimit : IsLimit (kernelCone f) := LinearMap.ext fun x => Subtype.ext_iff_val.2 (by simpa [← h] ) #align Module.kernel_is_limit ModuleCat.kernelIsLimit +/- warning: Module.cokernel_cocone -> ModuleCat.cokernelCocone is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.CokernelCofork.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) M N f +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.CokernelCofork.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u2, u1} R _inst_1)) M N f +Case conversion may be inaccurate. Consider using '#align Module.cokernel_cocone ModuleCat.cokernelCoconeₓ'. -/ /-- The cokernel cocone induced by the projection onto the quotient. -/ def cokernelCocone : CokernelCofork f := CokernelCofork.ofπ (asHom f.range.mkQ) <| LinearMap.range_mkQ_comp _ #align Module.cokernel_cocone ModuleCat.cokernelCocone +/- warning: Module.cokernel_is_colimit -> ModuleCat.cokernelIsColimit is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.IsColimit.{0, u2, 0, max u1 (succ u2)} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Limits.parallelPair.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) M N f (OfNat.ofNat.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (OfNat.mk.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (Zero.zero.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) M N))))) (ModuleCat.cokernelCocone.{u1, u2} R _inst_1 M N f) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u2, u1} R _inst_1} {N : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N), CategoryTheory.Limits.IsColimit.{0, u2, 0, max u1 (succ u2)} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Limits.parallelPair.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) M N f (OfNat.ofNat.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) 0 (Zero.toOfNat0.{u2} (Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) M N) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u2, u1} R _inst_1)) M N)))) (ModuleCat.cokernelCocone.{u1, u2} R _inst_1 M N f) +Case conversion may be inaccurate. Consider using '#align Module.cokernel_is_colimit ModuleCat.cokernelIsColimitₓ'. -/ /-- The projection onto the quotient is a cokernel in the categorical sense. -/ def cokernelIsColimit : IsColimit (cokernelCocone f) := Cofork.IsColimit.mk _ @@ -69,11 +93,23 @@ def cokernelIsColimit : IsColimit (cokernelCocone f) := end +/- warning: Module.has_kernels_Module -> ModuleCat.hasKernels_moduleCat is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], CategoryTheory.Limits.HasKernels.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) +but is expected to have type + forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R], CategoryTheory.Limits.HasKernels.{u1, max u2 (succ u1)} (ModuleCat.{u1, u2} R _inst_1) (ModuleCat.moduleCategory.{u1, u2} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, max u2 (succ u1)} (ModuleCat.{u1, u2} R _inst_1) (ModuleCat.moduleCategory.{u1, u2} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u1, u2} R _inst_1)) +Case conversion may be inaccurate. Consider using '#align Module.has_kernels_Module ModuleCat.hasKernels_moduleCatₓ'. -/ /-- The category of R-modules has kernels, given by the inclusion of the kernel submodule. -/ theorem hasKernels_moduleCat : HasKernels (ModuleCat R) := ⟨fun X Y f => HasLimit.mk ⟨_, kernelIsLimit f⟩⟩ #align Module.has_kernels_Module ModuleCat.hasKernels_moduleCat +/- warning: Module.has_cokernels_Module -> ModuleCat.hasCokernels_moduleCat is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], CategoryTheory.Limits.HasCokernels.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1) (ModuleCat.CategoryTheory.preadditive.{u2, u1} R _inst_1)) +but is expected to have type + forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R], CategoryTheory.Limits.HasCokernels.{u1, max u2 (succ u1)} (ModuleCat.{u1, u2} R _inst_1) (ModuleCat.moduleCategory.{u1, u2} R _inst_1) (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, max u2 (succ u1)} (ModuleCat.{u1, u2} R _inst_1) (ModuleCat.moduleCategory.{u1, u2} R _inst_1) (ModuleCat.instPreadditiveModuleCatModuleCategory.{u1, u2} R _inst_1)) +Case conversion may be inaccurate. Consider using '#align Module.has_cokernels_Module ModuleCat.hasCokernels_moduleCatₓ'. -/ /-- The category or R-modules has cokernels, given by the projection onto the quotient. -/ theorem hasCokernels_moduleCat : HasCokernels (ModuleCat R) := ⟨fun X Y f => HasColimit.mk ⟨_, cokernelIsColimit f⟩⟩ @@ -87,6 +123,12 @@ attribute [local instance] has_cokernels_Module variable {G H : ModuleCat.{v} R} (f : G ⟶ H) +/- warning: Module.kernel_iso_ker -> ModuleCat.kernelIsoKer is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {G : ModuleCat.{u2, u1} R _inst_1} {H : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) G H), CategoryTheory.Iso.{u2, max u1 (succ u2)} 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Consider using '#align Module.kernel_iso_ker_inv_kernel_ι ModuleCat.kernelIsoKer_inv_kernel_ιₓ'. -/ -- We now show this isomorphism commutes with the inclusion of the kernel into the source. @[simp, elementwise] theorem kernelIsoKer_inv_kernel_ι : (kernelIsoKer f).inv ≫ kernel.ι f = f.ker.Subtype := limit.isoLimitCone_inv_π _ _ #align Module.kernel_iso_ker_inv_kernel_ι ModuleCat.kernelIsoKer_inv_kernel_ι +/- warning: Module.kernel_iso_ker_hom_ker_subtype -> ModuleCat.kernelIsoKer_hom_ker_subtype is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {G : ModuleCat.{u2, u1} R _inst_1} {H : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) G H), Eq.{succ u2} 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Consider using '#align Module.kernel_iso_ker_hom_ker_subtype ModuleCat.kernelIsoKer_hom_ker_subtypeₓ'. -/ @[simp, elementwise] theorem kernelIsoKer_hom_ker_subtype : (kernelIsoKer f).hom ≫ f.ker.Subtype = kernel.ι f := IsLimit.conePointUniqueUpToIso_inv_comp _ (limit.isLimit _) WalkingParallelPair.zero #align Module.kernel_iso_ker_hom_ker_subtype ModuleCat.kernelIsoKer_hom_ker_subtype +/- warning: Module.cokernel_iso_range_quotient -> ModuleCat.cokernelIsoRangeQuotient is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {G : ModuleCat.{u2, u1} R _inst_1} {H : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (ModuleCat.moduleCategory.{u2, u1} R _inst_1))) G H), CategoryTheory.Iso.{u2, max u1 (succ 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Consider using '#align Module.cokernel_iso_range_quotient ModuleCat.cokernelIsoRangeQuotientₓ'. -/ /-- The categorical cokernel of a morphism in `Module` agrees with the usual module-theoretical quotient. -/ @@ -114,6 +174,12 @@ noncomputable def cokernelIsoRangeQuotient {G H : ModuleCat.{v} R} (f : G ⟶ H) colimit.isoColimitCocone ⟨_, cokernelIsColimit f⟩ #align Module.cokernel_iso_range_quotient ModuleCat.cokernelIsoRangeQuotient +/- warning: Module.cokernel_π_cokernel_iso_range_quotient_hom -> ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {G : ModuleCat.{u2, u1} R _inst_1} {H : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) 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Consider using '#align Module.cokernel_π_cokernel_iso_range_quotient_hom ModuleCat.cokernel_π_cokernelIsoRangeQuotient_homₓ'. -/ -- We now show this isomorphism commutes with the projection of target to the cokernel. @[simp, elementwise] theorem cokernel_π_cokernelIsoRangeQuotient_hom : @@ -121,12 +187,24 @@ theorem cokernel_π_cokernelIsoRangeQuotient_hom : convert colimit.iso_colimit_cocone_ι_hom _ _ <;> rfl #align Module.cokernel_π_cokernel_iso_range_quotient_hom ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom +/- warning: Module.range_mkq_cokernel_iso_range_quotient_inv -> ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {G : ModuleCat.{u2, u1} R _inst_1} {H : ModuleCat.{u2, u1} R _inst_1} (f : Quiver.Hom.{succ u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u1 (succ u2)} (ModuleCat.{u2, u1} R _inst_1) 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Consider using '#align Module.range_mkq_cokernel_iso_range_quotient_inv ModuleCat.range_mkQ_cokernelIsoRangeQuotient_invₓ'. -/ @[simp, elementwise] theorem range_mkQ_cokernelIsoRangeQuotient_inv : ↿f.range.mkQ ≫ (cokernelIsoRangeQuotient f).inv = cokernel.π f := by convert colimit.iso_colimit_cocone_ι_inv ⟨_, cokernel_is_colimit f⟩ _ <;> rfl #align Module.range_mkq_cokernel_iso_range_quotient_inv ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv +/- warning: Module.cokernel_π_ext -> ModuleCat.cokernel_π_ext is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : ModuleCat.{u1, u1} R _inst_1} {N : ModuleCat.{u1, u1} R _inst_1} (f : Quiver.Hom.{succ u1, succ u1} (ModuleCat.{u1, u1} R _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} (ModuleCat.{u1, u1} R _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} (ModuleCat.{u1, u1} R _inst_1) (ModuleCat.moduleCategory.{u1, u1} R _inst_1))) M N) {x : coeSort.{succ (succ u1), 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Consider using '#align Module.cokernel_π_ext ModuleCat.cokernel_π_extₓ'. -/ theorem cokernel_π_ext {M N : ModuleCat.{u} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m) : cokernel.π f x = cokernel.π f y := by subst w diff --git a/Mathbin/Algebra/Category/Mon/FilteredColimits.lean b/Mathbin/Algebra/Category/Mon/FilteredColimits.lean index 6e9543197e..65b95f7370 100644 --- a/Mathbin/Algebra/Category/Mon/FilteredColimits.lean +++ b/Mathbin/Algebra/Category/Mon/FilteredColimits.lean @@ -48,6 +48,7 @@ section -- passing around `F` all the time. parameter {J : Type v}[SmallCategory J](F : J ⥤ MonCat.{max v u}) +#print MonCat.FilteredColimits.M /- /-- The colimit of `F ⋙ forget Mon` in the category of types. In the following, we will construct a monoid structure on `M`. -/ @@ -57,7 +58,14 @@ abbrev M : Type max v u := Types.Quot (F ⋙ forget MonCat) #align Mon.filtered_colimits.M MonCat.FilteredColimits.M #align AddMon.filtered_colimits.M AddMonCat.FilteredColimits.M +-/ +/- warning: Mon.filtered_colimits.M.mk -> MonCat.FilteredColimits.M.mk is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}), (Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j))) -> (MonCat.FilteredColimits.M.{u1, u2} J _inst_1 F) +but is expected to have type + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1}), (Sigma.{u1, max u2 u1} J (fun (j : J) => CategoryTheory.Bundled.α.{max u2 u1, max u2 u1} Monoid.{max u2 u1} (Prefunctor.obj.{succ u1, max (succ u2) (succ u1), u1, max (succ u2) (succ u1)} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_1)) MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1} F) j))) -> (MonCat.FilteredColimits.M.{u1, u2} J _inst_1 F) +Case conversion may be inaccurate. Consider using '#align Mon.filtered_colimits.M.mk MonCat.FilteredColimits.M.mkₓ'. -/ /-- The canonical projection into the colimit, as a quotient type. -/ @[to_additive "The canonical projection into the colimit, as a quotient type."] abbrev M.mk : (Σj, F.obj j) → M := @@ -65,6 +73,12 @@ abbrev M.mk : (Σj, F.obj j) → M := #align Mon.filtered_colimits.M.mk MonCat.FilteredColimits.M.mk #align AddMon.filtered_colimits.M.mk AddMonCat.FilteredColimits.M.mk +/- warning: Mon.filtered_colimits.M.mk_eq -> MonCat.FilteredColimits.M.mk_eq is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) (x : Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, 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Consider using '#align Mon.filtered_colimits.M.mk_eq MonCat.FilteredColimits.M.mk_eqₓ'. -/ @[to_additive] theorem M.mk_eq (x y : Σj, F.obj j) (h : ∃ (k : J)(f : x.1 ⟶ k)(g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) : M.mk x = M.mk y := @@ -74,15 +88,23 @@ theorem M.mk_eq (x y : Σj, F.obj j) variable [IsFiltered J] +#print MonCat.FilteredColimits.colimitOne /- /-- As `J` is nonempty, we can pick an arbitrary object `j₀ : J`. We use this object to define the "one" in the colimit as the equivalence class of `⟨j₀, 1 : F.obj j₀⟩`. -/ @[to_additive "As `J` is nonempty, we can pick an arbitrary object `j₀ : J`. We use this object to\ndefine the \"zero\" in the colimit as the equivalence class of `⟨j₀, 0 : F.obj j₀⟩`."] -instance colimitHasOne : One M where one := M.mk ⟨IsFiltered.nonempty.some, 1⟩ -#align Mon.filtered_colimits.colimit_has_one MonCat.FilteredColimits.colimitHasOne -#align AddMon.filtered_colimits.colimit_has_zero AddMonCat.FilteredColimits.colimitHasZero +instance colimitOne : One M where one := M.mk ⟨IsFiltered.nonempty.some, 1⟩ +#align Mon.filtered_colimits.colimit_has_one MonCat.FilteredColimits.colimitOne +#align AddMon.filtered_colimits.colimit_has_zero AddMonCat.FilteredColimits.colimitZero +-/ +/- warning: Mon.filtered_colimits.colimit_one_eq -> MonCat.FilteredColimits.colimit_one_eq is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : 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Consider using '#align Mon.filtered_colimits.colimit_one_eq MonCat.FilteredColimits.colimit_one_eqₓ'. -/ /-- The definition of the "one" in the colimit is independent of the chosen object of `J`. In particular, this lemma allows us to "unfold" the definition of `colimit_one` at a custom chosen object `j`. @@ -97,6 +119,12 @@ theorem colimit_one_eq (j : J) : (1 : M) = M.mk ⟨j, 1⟩ := #align Mon.filtered_colimits.colimit_one_eq MonCat.FilteredColimits.colimit_one_eq #align AddMon.filtered_colimits.colimit_zero_eq AddMonCat.FilteredColimits.colimit_zero_eq +/- warning: Mon.filtered_colimits.colimit_mul_aux -> MonCat.FilteredColimits.colimitMulAux is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1], (Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j))) -> (Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j))) -> (MonCat.FilteredColimits.M.{u1, u2} J _inst_1 F) +but is expected to have type + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1], (Sigma.{u1, max u2 u1} J (fun (j : J) => CategoryTheory.Bundled.α.{max u2 u1, max u2 u1} Monoid.{max u2 u1} (Prefunctor.obj.{succ u1, max (succ u2) (succ u1), u1, max (succ u2) (succ u1)} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_1)) MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1} F) j))) -> (Sigma.{u1, max u2 u1} J (fun (j : J) => CategoryTheory.Bundled.α.{max u2 u1, max u2 u1} Monoid.{max u2 u1} (Prefunctor.obj.{succ u1, max (succ u2) (succ u1), u1, max (succ u2) (succ u1)} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_1)) MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1} F) j))) -> (MonCat.FilteredColimits.M.{u1, u2} J _inst_1 F) +Case conversion may be inaccurate. Consider using '#align Mon.filtered_colimits.colimit_mul_aux MonCat.FilteredColimits.colimitMulAuxₓ'. -/ /-- The "unlifted" version of multiplication in the colimit. To multiply two dependent pairs `⟨j₁, x⟩` and `⟨j₂, y⟩`, we pass to a common successor of `j₁` and `j₂` (given by `is_filtered.max`) and multiply them there. @@ -108,6 +136,12 @@ def colimitMulAux (x y : Σj, F.obj j) : M := #align Mon.filtered_colimits.colimit_mul_aux MonCat.FilteredColimits.colimitMulAux #align AddMon.filtered_colimits.colimit_add_aux AddMonCat.FilteredColimits.colimitAddAux +/- warning: Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_left -> MonCat.FilteredColimits.colimitMulAux_eq_of_rel_left is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1] {x : Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} 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Consider using '#align Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_left MonCat.FilteredColimits.colimitMulAux_eq_of_rel_leftₓ'. -/ /-- Multiplication in the colimit is well-defined in the left argument. -/ @[to_additive "Addition in the colimit is well-defined in the left argument."] theorem colimitMulAux_eq_of_rel_left {x x' y : Σj, F.obj j} @@ -124,8 +158,14 @@ theorem colimitMulAux_eq_of_rel_left {x x' y : Σj, F.obj j} dsimp simp_rw [MonoidHom.map_mul, ← comp_apply, ← F.map_comp, h₁, h₂, h₃, F.map_comp, comp_apply, hfg] #align Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_left MonCat.FilteredColimits.colimitMulAux_eq_of_rel_left -#align AddMon.filtered_colimits.colimit_add_aux_eq_of_rel_left AddMonCat.FilteredColimits.colimit_add_aux_eq_of_rel_left - +#align AddMon.filtered_colimits.colimit_add_aux_eq_of_rel_left AddMonCat.FilteredColimits.colimitAddAux_eq_of_rel_left + +/- warning: Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_right -> MonCat.FilteredColimits.colimitMulAux_eq_of_rel_right is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1] {x : Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j))} {y : Sigma.{u1, max u1 u2} J (fun (j : J) => coeSort.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} MonCat.{max u1 u2} Type.{max u1 u2} MonCat.hasCoeToSort.{max u1 u2} (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j))} {y' : Sigma.{u1, max u1 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Consider using '#align Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_right MonCat.FilteredColimits.colimitMulAux_eq_of_rel_rightₓ'. -/ /-- Multiplication in the colimit is well-defined in the right argument. -/ @[to_additive "Addition in the colimit is well-defined in the right argument."] theorem colimitMulAux_eq_of_rel_right {x y y' : Σj, F.obj j} @@ -142,11 +182,12 @@ theorem colimitMulAux_eq_of_rel_right {x y y' : Σj, F.obj j} dsimp simp_rw [MonoidHom.map_mul, ← comp_apply, ← F.map_comp, h₁, h₂, h₃, F.map_comp, comp_apply, hfg] #align Mon.filtered_colimits.colimit_mul_aux_eq_of_rel_right MonCat.FilteredColimits.colimitMulAux_eq_of_rel_right -#align AddMon.filtered_colimits.colimit_add_aux_eq_of_rel_right AddMonCat.FilteredColimits.colimit_add_aux_eq_of_rel_right +#align AddMon.filtered_colimits.colimit_add_aux_eq_of_rel_right AddMonCat.FilteredColimits.colimitAddAux_eq_of_rel_right +#print MonCat.FilteredColimits.colimitMul /- /-- Multiplication in the colimit. See also `colimit_mul_aux`. -/ @[to_additive "Addition in the colimit. See also `colimit_add_aux`."] -instance colimitHasMul : Mul M +instance colimitMul : Mul M where mul x y := by refine' Quot.lift₂ (colimit_mul_aux F) _ _ x y · intro x y y' h @@ -157,9 +198,16 @@ instance colimitHasMul : Mul M apply colimit_mul_aux_eq_of_rel_left apply types.filtered_colimit.rel_of_quot_rel exact h -#align Mon.filtered_colimits.colimit_has_mul MonCat.FilteredColimits.colimitHasMul -#align AddMon.filtered_colimits.colimit_has_add AddMonCat.FilteredColimits.colimitHasAdd +#align Mon.filtered_colimits.colimit_has_mul MonCat.FilteredColimits.colimitMul +#align AddMon.filtered_colimits.colimit_has_add AddMonCat.FilteredColimits.colimitAdd +-/ +/- warning: Mon.filtered_colimits.colimit_mul_mk_eq -> MonCat.FilteredColimits.colimit_mul_mk_eq is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} 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Consider using '#align Mon.filtered_colimits.colimit_mul_mk_eq MonCat.FilteredColimits.colimit_mul_mk_eqₓ'. -/ /-- Multiplication in the colimit is independent of the chosen "maximum" in the filtered category. In particular, this lemma allows us to "unfold" the definition of the multiplication of `x` and `y`, using a custom object `k` and morphisms `f : x.1 ⟶ k` and `g : y.1 ⟶ k`. @@ -178,6 +226,7 @@ theorem colimit_mul_mk_eq (x y : Σj, F.obj j) (k : J) (f : x.1 ⟶ k) (g : y.1 #align Mon.filtered_colimits.colimit_mul_mk_eq MonCat.FilteredColimits.colimit_mul_mk_eq #align AddMon.filtered_colimits.colimit_add_mk_eq AddMonCat.FilteredColimits.colimit_add_mk_eq +#print MonCat.FilteredColimits.colimitMonoid /- @[to_additive] instance colimitMonoid : Monoid M := { colimit_has_one, @@ -200,14 +249,23 @@ instance colimitMonoid : Monoid M := simp only [F.map_id, id_apply, mul_assoc] } #align Mon.filtered_colimits.colimit_monoid MonCat.FilteredColimits.colimitMonoid #align AddMon.filtered_colimits.colimit_add_monoid AddMonCat.FilteredColimits.colimitAddMonoid +-/ +#print MonCat.FilteredColimits.colimit /- /-- The bundled monoid giving the filtered colimit of a diagram. -/ @[to_additive "The bundled additive monoid giving the filtered colimit of a diagram."] def colimit : MonCat := MonCat.of M #align Mon.filtered_colimits.colimit MonCat.FilteredColimits.colimit #align AddMon.filtered_colimits.colimit AddMonCat.FilteredColimits.colimit +-/ +/- warning: Mon.filtered_colimits.cocone_morphism -> MonCat.FilteredColimits.coconeMorphism is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1] (j : J), Quiver.Hom.{succ (max u1 u2), succ (max u1 u2)} MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, succ (max u1 u2)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u1 u2, succ (max u1 u2)} MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2})) (CategoryTheory.Functor.obj.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2} F j) (MonCat.FilteredColimits.colimit.{u1, u2} J _inst_1 F _inst_2) +but is expected to have type + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1] (j : J), Quiver.Hom.{max (succ u2) (succ u1), max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1})) (Prefunctor.obj.{succ u1, max (succ u2) (succ u1), u1, max (succ u2) (succ u1)} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_1)) MonCat.{max u1 u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u1, max u2 u1, u1, max (succ u2) (succ u1)} J _inst_1 MonCat.{max u1 u2} instMonCatLargeCategory.{max u2 u1} F) j) (MonCat.FilteredColimits.colimit.{u1, u2} J _inst_1 F _inst_2) +Case conversion may be inaccurate. Consider using '#align Mon.filtered_colimits.cocone_morphism MonCat.FilteredColimits.coconeMorphismₓ'. -/ /-- The monoid homomorphism from a given monoid in the diagram to the colimit monoid. -/ @[to_additive "The additive monoid homomorphism from a given additive monoid in the diagram to the\ncolimit additive monoid."] @@ -221,6 +279,12 @@ def coconeMorphism (j : J) : F.obj j ⟶ colimit #align Mon.filtered_colimits.cocone_morphism MonCat.FilteredColimits.coconeMorphism #align AddMon.filtered_colimits.cocone_morphism AddMonCat.FilteredColimits.coconeMorphism +/- warning: Mon.filtered_colimits.cocone_naturality -> MonCat.FilteredColimits.cocone_naturality is a dubious translation: +lean 3 declaration is + forall {J : Type.{u1}} [_inst_1 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, max u1 u2, u1, succ (max u1 u2)} J _inst_1 MonCat.{max u1 u2} MonCat.largeCategory.{max u1 u2}) [_inst_2 : CategoryTheory.IsFiltered.{u1, u1} J _inst_1] {j : J} {j' : J} (f : 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Consider using '#align Mon.filtered_colimits.cocone_naturality MonCat.FilteredColimits.cocone_naturalityₓ'. -/ @[simp, to_additive] theorem cocone_naturality {j j' : J} (f : j ⟶ j') : F.map f ≫ cocone_morphism j' = cocone_morphism j := @@ -228,6 +292,7 @@ theorem cocone_naturality {j j' : J} (f : j ⟶ j') : #align Mon.filtered_colimits.cocone_naturality MonCat.FilteredColimits.cocone_naturality #align AddMon.filtered_colimits.cocone_naturality AddMonCat.FilteredColimits.cocone_naturality +#print MonCat.FilteredColimits.colimitCocone /- /-- The cocone over the proposed colimit monoid. -/ @[to_additive "The cocone over the proposed colimit additive monoid."] def colimitCocone : cocone F where @@ -235,7 +300,9 @@ def colimitCocone : cocone F where ι := { app := cocone_morphism } #align Mon.filtered_colimits.colimit_cocone MonCat.FilteredColimits.colimitCocone #align AddMon.filtered_colimits.colimit_cocone AddMonCat.FilteredColimits.colimitCocone +-/ +#print MonCat.FilteredColimits.colimitDesc /- /-- Given a cocone `t` of `F`, the induced monoid homomorphism from the colimit to the cocone point. As a function, this is simply given by the induced map of the corresponding cocone in `Type`. The only thing left to see is that it is a monoid homomorphism. @@ -256,7 +323,9 @@ def colimitDesc (t : cocone F) : colimit ⟶ t.pt rw [MonoidHom.map_mul, t.w_apply, t.w_apply] #align Mon.filtered_colimits.colimit_desc MonCat.FilteredColimits.colimitDesc #align AddMon.filtered_colimits.colimit_desc AddMonCat.FilteredColimits.colimitDesc +-/ +#print MonCat.FilteredColimits.colimitCoconeIsColimit /- /-- The proposed colimit cocone is a colimit in `Mon`. -/ @[to_additive "The proposed colimit cocone is a colimit in `AddMon`."] def colimitCoconeIsColimit : IsColimit colimit_cocone @@ -271,7 +340,9 @@ def colimitCoconeIsColimit : IsColimit colimit_cocone fun j => funext fun x => MonoidHom.congr_fun (h j) x #align Mon.filtered_colimits.colimit_cocone_is_colimit MonCat.FilteredColimits.colimitCoconeIsColimit #align AddMon.filtered_colimits.colimit_cocone_is_colimit AddMonCat.FilteredColimits.colimitCoconeIsColimit +-/ +#print MonCat.FilteredColimits.forgetPreservesFilteredColimits /- @[to_additive] instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget MonCat.{u}) where PreservesFilteredColimits J _ _ := @@ -281,6 +352,7 @@ instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget Mon (types.colimit_cocone_is_colimit (F ⋙ forget MonCat.{u})) } #align Mon.filtered_colimits.forget_preserves_filtered_colimits MonCat.FilteredColimits.forgetPreservesFilteredColimits #align AddMon.filtered_colimits.forget_preserves_filtered_colimits AddMonCat.FilteredColimits.forgetPreservesFilteredColimits +-/ end @@ -296,16 +368,19 @@ section -- passing around `F` all the time. parameter {J : Type v}[SmallCategory J][IsFiltered J](F : J ⥤ CommMonCat.{max v u}) +#print CommMonCat.FilteredColimits.M /- /-- The colimit of `F ⋙ forget₂ CommMon Mon` in the category `Mon`. In the following, we will show that this has the structure of a _commutative_ monoid. -/ @[to_additive "The colimit of `F ⋙ forget₂ AddCommMon AddMon` in the category `AddMon`. In the\nfollowing, we will show that this has the structure of a _commutative_ additive monoid."] -abbrev m : MonCat := +abbrev M : MonCat := MonCat.FilteredColimits.colimit (F ⋙ forget₂ CommMonCat MonCat.{max v u}) -#align CommMon.filtered_colimits.M CommMonCat.FilteredColimits.m -#align AddCommMon.filtered_colimits.M AddCommMonCat.FilteredColimits.m +#align CommMon.filtered_colimits.M CommMonCat.FilteredColimits.M +#align AddCommMon.filtered_colimits.M AddCommMonCat.FilteredColimits.M +-/ +#print CommMonCat.FilteredColimits.colimitCommMonoid /- @[to_additive] instance colimitCommMonoid : CommMonoid M := { M.Monoid with @@ -319,14 +394,18 @@ instance colimitCommMonoid : CommMonoid M := rw [mul_comm] } #align CommMon.filtered_colimits.colimit_comm_monoid CommMonCat.FilteredColimits.colimitCommMonoid #align AddCommMon.filtered_colimits.colimit_add_comm_monoid AddCommMonCat.FilteredColimits.colimitAddCommMonoid +-/ +#print CommMonCat.FilteredColimits.colimit /- /-- The bundled commutative monoid giving the filtered colimit of a diagram. -/ @[to_additive "The bundled additive commutative monoid giving the filtered colimit of a diagram."] def colimit : CommMonCat := CommMonCat.of M #align CommMon.filtered_colimits.colimit CommMonCat.FilteredColimits.colimit #align AddCommMon.filtered_colimits.colimit AddCommMonCat.FilteredColimits.colimit +-/ +#print CommMonCat.FilteredColimits.colimitCocone /- /-- The cocone over the proposed colimit commutative monoid. -/ @[to_additive "The cocone over the proposed colimit additive commutative monoid."] def colimitCocone : cocone F where @@ -334,7 +413,9 @@ def colimitCocone : cocone F where ι := { (MonCat.FilteredColimits.colimitCocone (F ⋙ forget₂ CommMonCat MonCat.{max v u})).ι with } #align CommMon.filtered_colimits.colimit_cocone CommMonCat.FilteredColimits.colimitCocone #align AddCommMon.filtered_colimits.colimit_cocone AddCommMonCat.FilteredColimits.colimitCocone +-/ +#print CommMonCat.FilteredColimits.colimitCoconeIsColimit /- /-- The proposed colimit cocone is a colimit in `CommMon`. -/ @[to_additive "The proposed colimit cocone is a colimit in `AddCommMon`."] def colimitCoconeIsColimit : IsColimit colimit_cocone @@ -351,7 +432,9 @@ def colimitCoconeIsColimit : IsColimit colimit_cocone m fun j => funext fun x => MonoidHom.congr_fun (h j) x #align CommMon.filtered_colimits.colimit_cocone_is_colimit CommMonCat.FilteredColimits.colimitCoconeIsColimit #align AddCommMon.filtered_colimits.colimit_cocone_is_colimit AddCommMonCat.FilteredColimits.colimitCoconeIsColimit +-/ +#print CommMonCat.FilteredColimits.forget₂MonPreservesFilteredColimits /- @[to_additive forget₂_AddMon_preserves_filtered_colimits] instance forget₂MonPreservesFilteredColimits : PreservesFilteredColimits (forget₂ CommMonCat MonCat.{u}) @@ -362,12 +445,15 @@ instance forget₂MonPreservesFilteredColimits : (MonCat.FilteredColimits.colimitCoconeIsColimit (F ⋙ forget₂ CommMonCat MonCat.{u})) } #align CommMon.filtered_colimits.forget₂_Mon_preserves_filtered_colimits CommMonCat.FilteredColimits.forget₂MonPreservesFilteredColimits #align AddCommMon.filtered_colimits.forget₂_AddMon_preserves_filtered_colimits AddCommMonCat.FilteredColimits.forget₂AddMonPreservesFilteredColimits +-/ +#print CommMonCat.FilteredColimits.forgetPreservesFilteredColimits /- @[to_additive] instance forgetPreservesFilteredColimits : PreservesFilteredColimits (forget CommMonCat.{u}) := Limits.compPreservesFilteredColimits (forget₂ CommMonCat MonCat) (forget MonCat) #align CommMon.filtered_colimits.forget_preserves_filtered_colimits CommMonCat.FilteredColimits.forgetPreservesFilteredColimits #align AddCommMon.filtered_colimits.forget_preserves_filtered_colimits AddCommMonCat.FilteredColimits.forgetPreservesFilteredColimits +-/ end diff --git a/Mathbin/AlgebraicGeometry/AffineScheme.lean b/Mathbin/AlgebraicGeometry/AffineScheme.lean index 0ebd274f47..4d6e4c2bfe 100644 --- a/Mathbin/AlgebraicGeometry/AffineScheme.lean +++ b/Mathbin/AlgebraicGeometry/AffineScheme.lean @@ -162,20 +162,20 @@ def Scheme.affineOpens (X : Scheme) : Set (Opens X.carrier) := { U : Opens X.carrier | IsAffineOpen U } #align algebraic_geometry.Scheme.affine_opens AlgebraicGeometry.Scheme.affineOpens -theorem range_isAffineOpen_of_open_immersion {X Y : Scheme} [IsAffine X] (f : X ⟶ Y) +theorem rangeIsAffineOpenOfOpenImmersion {X Y : Scheme} [IsAffine X] (f : X ⟶ Y) [H : IsOpenImmersion f] : IsAffineOpen f.opensRange := by refine' is_affine_of_iso (is_open_immersion.iso_of_range_eq f (Y.of_restrict _) _).inv exact subtype.range_coe.symm infer_instance -#align algebraic_geometry.range_is_affine_open_of_open_immersion AlgebraicGeometry.range_isAffineOpen_of_open_immersion +#align algebraic_geometry.range_is_affine_open_of_open_immersion AlgebraicGeometry.rangeIsAffineOpenOfOpenImmersion -theorem top_isAffineOpen (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : Opens X.carrier) := +theorem topIsAffineOpen (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : Opens X.carrier) := by convert range_is_affine_open_of_open_immersion (𝟙 X) ext1 exact set.range_id.symm -#align algebraic_geometry.top_is_affine_open AlgebraicGeometry.top_isAffineOpen +#align algebraic_geometry.top_is_affine_open AlgebraicGeometry.topIsAffineOpen instance Scheme.affineCover_isAffine (X : Scheme) (i : X.affineCover.J) : IsAffine (X.affineCover.obj i) := @@ -245,20 +245,19 @@ theorem IsAffineOpen.isCompact {X : Scheme} {U : Opens X.carrier} (hU : IsAffine exact Set.image_univ #align algebraic_geometry.is_affine_open.is_compact AlgebraicGeometry.IsAffineOpen.isCompact -theorem IsAffineOpen.image_isOpenImmersion {X Y : Scheme} {U : Opens X.carrier} - (hU : IsAffineOpen U) (f : X ⟶ Y) [H : IsOpenImmersion f] : - IsAffineOpen (f.opensFunctor.obj U) := +theorem IsAffineOpen.imageIsOpenImmersion {X Y : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) + (f : X ⟶ Y) [H : IsOpenImmersion f] : IsAffineOpen (f.opensFunctor.obj U) := by haveI : is_affine _ := hU convert range_is_affine_open_of_open_immersion (X.of_restrict U.open_embedding ≫ f) ext1 exact Set.image_eq_range _ _ -#align algebraic_geometry.is_affine_open.image_is_open_immersion AlgebraicGeometry.IsAffineOpen.image_isOpenImmersion +#align algebraic_geometry.is_affine_open.image_is_open_immersion AlgebraicGeometry.IsAffineOpen.imageIsOpenImmersion theorem isAffineOpen_iff_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : Opens X.carrier) : IsAffineOpen (H.openFunctor.obj U) ↔ IsAffineOpen U := by - refine' ⟨fun hU => @is_affine_of_iso _ _ hU, fun hU => hU.image_isOpenImmersion f⟩ + refine' ⟨fun hU => @is_affine_of_iso _ _ hU, fun hU => hU.imageIsOpenImmersion f⟩ refine' (is_open_immersion.iso_of_range_eq (X.of_restrict _ ≫ f) (Y.of_restrict _) _).Hom · rw [Scheme.comp_val_base, coe_comp, Set.range_comp] dsimp [opens.inclusion] @@ -268,7 +267,7 @@ theorem isAffineOpen_iff_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : Is #align algebraic_geometry.is_affine_open_iff_of_is_open_immersion AlgebraicGeometry.isAffineOpen_iff_of_isOpenImmersion instance Scheme.quasi_compact_of_affine (X : Scheme) [IsAffine X] : CompactSpace X.carrier := - ⟨(top_isAffineOpen X).IsCompact⟩ + ⟨(topIsAffineOpen X).IsCompact⟩ #align algebraic_geometry.Scheme.quasi_compact_of_affine AlgebraicGeometry.Scheme.quasi_compact_of_affine theorem IsAffineOpen.fromSpec_base_preimage {X : Scheme} {U : Opens X.carrier} @@ -336,7 +335,7 @@ theorem IsAffineOpen.fromSpec_app_eq {X : Scheme} {U : Opens X.carrier} (hU : Is by rw [← hU.Spec_Γ_identity_hom_app_from_Spec, iso.inv_hom_id_app_assoc] #align algebraic_geometry.is_affine_open.from_Spec_app_eq AlgebraicGeometry.IsAffineOpen.fromSpec_app_eq -theorem IsAffineOpen.basicOpen_is_affine {X : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) +theorem IsAffineOpen.basicOpenIsAffine {X : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) (f : X.Presheaf.obj (op U)) : IsAffineOpen (X.basicOpen f) := by convert range_is_affine_open_of_open_immersion @@ -371,9 +370,9 @@ theorem IsAffineOpen.basicOpen_is_affine {X : Scheme} {U : Opens X.carrier} (hU congr 2 rw [iso.eq_inv_comp] erw [hU.Spec_Γ_identity_hom_app_from_Spec] -#align algebraic_geometry.is_affine_open.basic_open_is_affine AlgebraicGeometry.IsAffineOpen.basicOpen_is_affine +#align algebraic_geometry.is_affine_open.basic_open_is_affine AlgebraicGeometry.IsAffineOpen.basicOpenIsAffine -theorem IsAffineOpen.map_restrict_basicOpen {X : Scheme} (r : X.Presheaf.obj (op ⊤)) +theorem IsAffineOpen.mapRestrictBasicOpen {X : Scheme} (r : X.Presheaf.obj (op ⊤)) {U : Opens X.carrier} (hU : IsAffineOpen U) : IsAffineOpen ((Opens.map (X.of_restrict (X.basicOpen r).OpenEmbedding).1.base).obj U) := by @@ -384,7 +383,7 @@ theorem IsAffineOpen.map_restrict_basicOpen {X : Scheme} (r : X.Presheaf.obj (op erw [opens.functor_obj_map_obj, opens.open_embedding_obj_top, inf_comm, ← Scheme.basic_open_res _ _ (hom_of_le le_top).op] exact hU.basic_open_is_affine _ -#align algebraic_geometry.is_affine_open.map_restrict_basic_open AlgebraicGeometry.IsAffineOpen.map_restrict_basicOpen +#align algebraic_geometry.is_affine_open.map_restrict_basic_open AlgebraicGeometry.IsAffineOpen.mapRestrictBasicOpen theorem Scheme.map_prime_spectrum_basicOpen_of_affine (X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) : @@ -519,7 +518,7 @@ theorem is_localization_basicOpen {X : Scheme} {U : Opens X.carrier} (hU : IsAff instance {X : Scheme} [IsAffine X] (r : X.Presheaf.obj (op ⊤)) : IsLocalization.Away r (X.Presheaf.obj (op <| X.basicOpen r)) := - is_localization_basicOpen (top_isAffineOpen X) r + is_localization_basicOpen (topIsAffineOpen X) r theorem is_localization_of_eq_basicOpen {X : Scheme} {U V : Opens X.carrier} (i : V ⟶ U) (hU : IsAffineOpen U) (r : X.Presheaf.obj (op U)) (e : V = X.basicOpen r) : @@ -536,7 +535,7 @@ instance ΓRestrictAlgebra {X : Scheme} {Y : TopCat} {f : Y ⟶ X.carrier} (hf : instance Γ_restrict_is_localization (X : Scheme.{u}) [IsAffine X] (r : Scheme.Γ.obj (op X)) : IsLocalization.Away r (Scheme.Γ.obj (op <| X.restrict (X.basicOpen r).OpenEmbedding)) := - is_localization_of_eq_basicOpen _ (top_isAffineOpen X) r (Opens.openEmbedding_obj_top _) + is_localization_of_eq_basicOpen _ (topIsAffineOpen X) r (Opens.openEmbedding_obj_top _) #align algebraic_geometry.Γ_restrict_is_localization AlgebraicGeometry.Γ_restrict_is_localization theorem basicOpen_basicOpen_is_basicOpen {X : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) @@ -673,7 +672,7 @@ theorem IsAffineOpen.is_localization_stalk {X : Scheme} {U : Opens X.carrier} (h @[simps] def Scheme.affineBasicOpen (X : Scheme) {U : X.affineOpens} (f : X.Presheaf.obj <| op U) : X.affineOpens := - ⟨X.basicOpen f, U.Prop.basicOpen_is_affine f⟩ + ⟨X.basicOpen f, U.Prop.basicOpenIsAffine f⟩ #align algebraic_geometry.Scheme.affine_basic_open AlgebraicGeometry.Scheme.affineBasicOpen @[simp] diff --git a/Mathbin/AlgebraicGeometry/Gluing.lean b/Mathbin/AlgebraicGeometry/Gluing.lean index e20509068a..8bb814b0aa 100644 --- a/Mathbin/AlgebraicGeometry/Gluing.lean +++ b/Mathbin/AlgebraicGeometry/Gluing.lean @@ -438,7 +438,7 @@ instance : Epi 𝒰.fromGlued.val.base := exact h instance fromGlued_open_immersion : IsOpenImmersion 𝒰.fromGlued := - SheafedSpace.IsOpenImmersion.of_stalk_iso _ 𝒰.fromGlued_openEmbedding + SheafedSpace.IsOpenImmersion.ofStalkIso _ 𝒰.fromGlued_openEmbedding #align algebraic_geometry.Scheme.open_cover.from_glued_open_immersion AlgebraicGeometry.Scheme.OpenCover.fromGlued_open_immersion instance : IsIso 𝒰.fromGlued := diff --git a/Mathbin/AlgebraicGeometry/Limits.lean b/Mathbin/AlgebraicGeometry/Limits.lean index 443e922e73..26cf487737 100644 --- a/Mathbin/AlgebraicGeometry/Limits.lean +++ b/Mathbin/AlgebraicGeometry/Limits.lean @@ -139,12 +139,12 @@ instance initial_isEmpty : IsEmpty (⊥_ Scheme).carrier := ⟨fun x => ((initial.to Scheme.empty : _).1.base x).elim⟩ #align algebraic_geometry.initial_is_empty AlgebraicGeometry.initial_isEmpty -theorem bot_isAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := +theorem botIsAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := by convert range_is_affine_open_of_open_immersion (initial.to X) ext exact (false_iff_iff _).mpr fun x => isEmptyElim (show (⊥_ Scheme).carrier from x.some) -#align algebraic_geometry.bot_is_affine_open AlgebraicGeometry.bot_isAffineOpen +#align algebraic_geometry.bot_is_affine_open AlgebraicGeometry.botIsAffineOpen instance : HasStrictInitialObjects Scheme := hasStrictInitialObjects_of_initial_is_strict fun A f => by infer_instance diff --git a/Mathbin/AlgebraicGeometry/Morphisms/Basic.lean b/Mathbin/AlgebraicGeometry/Morphisms/Basic.lean index 168b9bbaf9..6953a4d259 100644 --- a/Mathbin/AlgebraicGeometry/Morphisms/Basic.lean +++ b/Mathbin/AlgebraicGeometry/Morphisms/Basic.lean @@ -129,11 +129,11 @@ def targetAffineLocally (P : AffineTargetMorphismProperty) : MorphismProperty Sc fun {X Y : Scheme} (f : X ⟶ Y) => ∀ U : Y.affineOpens, @P (f ∣_ U) U.Prop #align algebraic_geometry.target_affine_locally AlgebraicGeometry.targetAffineLocally -theorem IsAffineOpen.map_isIso {X Y : Scheme} {U : Opens Y.carrier} (hU : IsAffineOpen U) - (f : X ⟶ Y) [IsIso f] : IsAffineOpen ((Opens.map f.1.base).obj U) := +theorem IsAffineOpen.mapIsIso {X Y : Scheme} {U : Opens Y.carrier} (hU : IsAffineOpen U) (f : X ⟶ Y) + [IsIso f] : IsAffineOpen ((Opens.map f.1.base).obj U) := haveI : is_affine _ := hU is_affine_of_iso (f ∣_ U) -#align algebraic_geometry.is_affine_open.map_is_iso AlgebraicGeometry.IsAffineOpen.map_isIso +#align algebraic_geometry.is_affine_open.map_is_iso AlgebraicGeometry.IsAffineOpen.mapIsIso theorem targetAffineLocally_respectsIso {P : AffineTargetMorphismProperty} (hP : P.toProperty.RespectsIso) : (targetAffineLocally P).RespectsIso := @@ -162,11 +162,11 @@ structure AffineTargetMorphismProperty.IsLocal (P : AffineTargetMorphismProperty RespectsIso : P.toProperty.RespectsIso toBasicOpen : ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (r : Y.Presheaf.obj <| op ⊤), - P f → @P (f ∣_ Y.basic_open r) ((top_is_affine_open Y).basicOpen_is_affine _) + P f → @P (f ∣_ Y.basic_open r) ((top_is_affine_open Y).basicOpenIsAffine _) ofBasicOpenCover : ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (s : Finset (Y.Presheaf.obj <| op ⊤)) (hs : Ideal.span (s : Set (Y.Presheaf.obj <| op ⊤)) = ⊤), - (∀ r : s, @P (f ∣_ Y.basic_open r.1) ((top_is_affine_open Y).basicOpen_is_affine _)) → P f + (∀ r : s, @P (f ∣_ Y.basic_open r.1) ((top_is_affine_open Y).basicOpenIsAffine _)) → P f #align algebraic_geometry.affine_target_morphism_property.is_local AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal theorem targetAffineLocallyOfOpenCover {P : AffineTargetMorphismProperty} (hP : P.IsLocal) @@ -277,7 +277,7 @@ theorem AffineTargetMorphismProperty.isLocalOfOpenCoverImply (P : AffineTargetMo refine' ⟨hP, _, _⟩ · introv h skip - haveI : is_affine _ := (top_is_affine_open Y).basicOpen_is_affine r + haveI : is_affine _ := (top_is_affine_open Y).basicOpenIsAffine r delta morphism_restrict rw [affine_cancel_left_is_iso hP] refine' @H f ⟨Scheme.open_cover_of_is_iso (𝟙 Y), _, _⟩ (Y.of_restrict _) _inst _ @@ -294,10 +294,10 @@ theorem AffineTargetMorphismProperty.isLocalOfOpenCoverImply (P : AffineTargetMo rwa [← category.comp_id pullback.snd, ← pullback.condition, affine_cancel_left_is_iso hP] at this · intro i - exact (top_is_affine_open Y).basicOpen_is_affine _ + exact (top_is_affine_open Y).basicOpenIsAffine _ · rintro (i : s) specialize hs' i - haveI : is_affine _ := (top_is_affine_open Y).basicOpen_is_affine i.1 + haveI : is_affine _ := (top_is_affine_open Y).basicOpenIsAffine i.1 delta morphism_restrict at hs' rwa [affine_cancel_left_is_iso hP] at hs' #align algebraic_geometry.affine_target_morphism_property.is_local_of_open_cover_imply AlgebraicGeometry.AffineTargetMorphismProperty.isLocalOfOpenCoverImply diff --git a/Mathbin/AlgebraicGeometry/Morphisms/QuasiCompact.lean b/Mathbin/AlgebraicGeometry/Morphisms/QuasiCompact.lean index 12349704d9..872c5505cc 100644 --- a/Mathbin/AlgebraicGeometry/Morphisms/QuasiCompact.lean +++ b/Mathbin/AlgebraicGeometry/Morphisms/QuasiCompact.lean @@ -91,7 +91,7 @@ theorem isCompact_open_iff_eq_basicOpen_union {X : Scheme} [IsAffine X] (U : Set ∃ s : Set (X.Presheaf.obj (op ⊤)), s.Finite ∧ U = ⋃ (i : X.Presheaf.obj (op ⊤)) (h : i ∈ s), X.basicOpen i := (isBasis_basicOpen X).isCompact_open_iff_eq_finite_iUnion _ - (fun i => ((top_isAffineOpen _).basicOpen_is_affine _).IsCompact) _ + (fun i => ((topIsAffineOpen _).basicOpenIsAffine _).IsCompact) _ #align algebraic_geometry.is_compact_open_iff_eq_basic_open_union AlgebraicGeometry.isCompact_open_iff_eq_basicOpen_union theorem quasiCompact_iff_forall_affine : diff --git a/Mathbin/AlgebraicGeometry/Morphisms/QuasiSeparated.lean b/Mathbin/AlgebraicGeometry/Morphisms/QuasiSeparated.lean index 803dd67eff..c2d84fb4b5 100644 --- a/Mathbin/AlgebraicGeometry/Morphisms/QuasiSeparated.lean +++ b/Mathbin/AlgebraicGeometry/Morphisms/QuasiSeparated.lean @@ -277,7 +277,7 @@ instance quasiSeparatedSpace_of_isAffine (X : Scheme) [IsAffine X] : intro i' hi' change IsCompact (X.basic_open i ⊓ X.basic_open i').1 rw [← Scheme.basic_open_mul] - exact ((top_is_affine_open _).basicOpen_is_affine _).IsCompact + exact ((top_is_affine_open _).basicOpenIsAffine _).IsCompact #align algebraic_geometry.quasi_separated_space_of_is_affine AlgebraicGeometry.quasiSeparatedSpace_of_isAffine theorem IsAffineOpen.isQuasiSeparated {X : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) : diff --git a/Mathbin/AlgebraicGeometry/Morphisms/RingHomProperties.lean b/Mathbin/AlgebraicGeometry/Morphisms/RingHomProperties.lean index 108a1e2a72..8864de8f7f 100644 --- a/Mathbin/AlgebraicGeometry/Morphisms/RingHomProperties.lean +++ b/Mathbin/AlgebraicGeometry/Morphisms/RingHomProperties.lean @@ -217,7 +217,7 @@ theorem affineLocally_iff_affineOpens_le (hP : RingHom.RespectsIso @P) {X Y : Sc convert V.2 infer_instance · intro H V - specialize H ⟨_, V.2.image_isOpenImmersion (X.of_restrict _)⟩ (Subtype.coe_image_subset _ _) + specialize H ⟨_, V.2.imageIsOpenImmersion (X.of_restrict _)⟩ (Subtype.coe_image_subset _ _) erw [← X.presheaf.map_comp] rw [← hP.cancel_right_is_iso _ (X.presheaf.map (eq_to_hom _)), category.assoc, ← X.presheaf.map_comp] @@ -238,12 +238,12 @@ theorem scheme_restrict_basicOpen_of_localizationPreserves (h₁ : RingHom.Respe U.1.OpenEmbedding ≫ f ∣_ Y.basicOpen r).op) := by - specialize H ⟨_, U.2.image_isOpenImmersion (X.of_restrict _)⟩ + specialize H ⟨_, U.2.imageIsOpenImmersion (X.of_restrict _)⟩ convert(h₁.of_restrict_morphism_restrict_iff _ _ _ _ _).mpr _ using 1 pick_goal 5 · exact h₂.away r H · infer_instance - · exact U.2.image_isOpenImmersion _ + · exact U.2.imageIsOpenImmersion _ · ext1 exact (Set.preimage_image_eq _ Subtype.coe_injective).symm #align algebraic_geometry.Scheme_restrict_basic_open_of_localization_preserves AlgebraicGeometry.scheme_restrict_basicOpen_of_localizationPreserves @@ -272,7 +272,7 @@ theorem sourceAffineLocallyIsLocal (h₁ : RingHom.RespectsIso @P) intro V hV rw [Scheme.preimage_basic_open] at hV subst hV - exact U.2.map_restrict_basicOpen (Scheme.Γ.map f.op r.1) + exact U.2.mapRestrictBasicOpen (Scheme.Γ.map f.op r.1) #align algebraic_geometry.source_affine_locally_is_local AlgebraicGeometry.sourceAffineLocallyIsLocal variable {P} (hP : RingHom.PropertyIsLocal @P) diff --git a/Mathbin/AlgebraicGeometry/OpenImmersion.lean b/Mathbin/AlgebraicGeometry/OpenImmersion.lean index 9bc8589276..490eebeb1c 100644 --- a/Mathbin/AlgebraicGeometry/OpenImmersion.lean +++ b/Mathbin/AlgebraicGeometry/OpenImmersion.lean @@ -636,9 +636,9 @@ theorem toSheafedSpaceHom_c : (toSheafedSpaceHom Y f).c = f.c := rfl #align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom_c AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom_c -instance toSheafedSpace_isOpenImmersion : SheafedSpace.IsOpenImmersion (toSheafedSpaceHom Y f) := +instance toSheafedSpaceIsOpenImmersion : SheafedSpace.IsOpenImmersion (toSheafedSpaceHom Y f) := H -#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace_isOpenImmersion +#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceIsOpenImmersion omit H @@ -719,10 +719,10 @@ end PresheafedSpace.IsOpenImmersion namespace SheafedSpace.IsOpenImmersion -instance (priority := 100) of_isIso {X Y : SheafedSpace.{v} C} (f : X ⟶ Y) [IsIso f] : +instance (priority := 100) ofIsIso {X Y : SheafedSpace.{v} C} (f : X ⟶ Y) [IsIso f] : SheafedSpace.IsOpenImmersion f := @PresheafedSpace.IsOpenImmersion.ofIsIso _ f (SheafedSpace.forgetToPresheafedSpace.map_isIso _) -#align algebraic_geometry.SheafedSpace.is_open_immersion.of_is_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_isIso +#align algebraic_geometry.SheafedSpace.is_open_immersion.of_is_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofIsIso instance comp {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) [SheafedSpace.IsOpenImmersion f] [SheafedSpace.IsOpenImmersion g] : SheafedSpace.IsOpenImmersion (f ≫ g) := @@ -811,7 +811,7 @@ instance sheafedSpace_hasPullback_of_right : HasPullback g f := #align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_has_pullback_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_hasPullback_of_right /-- Open immersions are stable under base-change. -/ -instance sheafedSpace_pullback_snd_of_left : +instance sheafedSpacePullbackSndOfLeft : SheafedSpace.IsOpenImmersion (pullback.snd : pullback f g ⟶ _) := by delta pullback.snd @@ -822,9 +822,9 @@ instance sheafedSpace_pullback_snd_of_left : rw [← this] dsimp infer_instance -#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_snd_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_snd_of_left +#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_snd_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpacePullbackSndOfLeft -instance sheafedSpace_pullback_fst_of_right : +instance sheafedSpacePullbackFstOfRight : SheafedSpace.IsOpenImmersion (pullback.fst : pullback g f ⟶ _) := by delta pullback.fst @@ -835,14 +835,14 @@ instance sheafedSpace_pullback_fst_of_right : rw [← this] dsimp infer_instance -#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_fst_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_fst_of_right +#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_fst_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpacePullbackFstOfRight -instance sheafedSpace_pullback_to_base_isOpenImmersion [SheafedSpace.IsOpenImmersion g] : +instance sheafedSpacePullbackToBaseIsOpenImmersion [SheafedSpace.IsOpenImmersion g] : SheafedSpace.IsOpenImmersion (limit.π (cospan f g) one : pullback f g ⟶ Z) := by rw [← limit.w (cospan f g) hom.inl, cospan_map_inl] infer_instance -#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_to_base_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_to_base_isOpenImmersion +#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_to_base_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpacePullbackToBaseIsOpenImmersion end Pullback @@ -859,7 +859,7 @@ whose forgetful functor reflects isomorphisms, preserves limits and filtered col Then a morphism `X ⟶ Y` that is a topological open embedding is an open immersion iff every stalk map is an iso. -/ -theorem of_stalk_iso {X Y : SheafedSpace C} (f : X ⟶ Y) (hf : OpenEmbedding f.base) +theorem ofStalkIso {X Y : SheafedSpace C} (f : X ⟶ Y) (hf : OpenEmbedding f.base) [H : ∀ x : X, IsIso (PresheafedSpace.stalkMap f x)] : SheafedSpace.IsOpenImmersion f := { base_open := hf c_iso := fun U => @@ -875,7 +875,7 @@ theorem of_stalk_iso {X Y : SheafedSpace C} (f : X ⟶ Y) (hf : OpenEmbedding f. have := @is_iso.comp_is_iso _ H (@is_iso.inv_is_iso _ H') rw [category.assoc, is_iso.hom_inv_id, category.comp_id] at this exact this } -#align algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso +#align algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofStalkIso end OfStalkIso @@ -929,7 +929,7 @@ theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.ob exact h (congr_arg discrete.mk (congr_arg Sigma.fst Eq)) #align algebraic_geometry.SheafedSpace.is_open_immersion.image_preimage_is_empty AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty -instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] : +instance sigmaιIsOpenImmersion [HasStrictTerminalObjects C] : SheafedSpace.IsOpenImmersion (colimit.ι F i) where base_open := sigma_ι_openEmbedding F i @@ -966,7 +966,7 @@ instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] : convert(F.obj j).Sheaf.isTerminalOfEmpty convert image_preimage_is_empty F i j (fun h => hj (congr_arg op h.symm)) U exact (congr_arg PresheafedSpace.hom.base e).symm -#align algebraic_geometry.SheafedSpace.is_open_immersion.sigma_ι_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenImmersion +#align algebraic_geometry.SheafedSpace.is_open_immersion.sigma_ι_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigmaιIsOpenImmersion end Prod @@ -1627,7 +1627,7 @@ theorem to_iso {X Y : Scheme} (f : X ⟶ Y) [h : IsOpenImmersion f] [Epi f.1.bas theorem of_stalk_iso {X Y : Scheme} (f : X ⟶ Y) (hf : OpenEmbedding f.1.base) [∀ x, IsIso (PresheafedSpace.stalkMap f.1 x)] : IsOpenImmersion f := - SheafedSpace.IsOpenImmersion.of_stalk_iso f.1 hf + SheafedSpace.IsOpenImmersion.ofStalkIso f.1 hf #align algebraic_geometry.is_open_immersion.of_stalk_iso AlgebraicGeometry.IsOpenImmersion.of_stalk_iso theorem iff_stalk_iso {X Y : Scheme} (f : X ⟶ Y) : diff --git a/Mathbin/AlgebraicGeometry/PresheafedSpace.lean b/Mathbin/AlgebraicGeometry/PresheafedSpace.lean index feda2701bb..1b3e0740bf 100644 --- a/Mathbin/AlgebraicGeometry/PresheafedSpace.lean +++ b/Mathbin/AlgebraicGeometry/PresheafedSpace.lean @@ -40,6 +40,12 @@ attribute [local tidy] tactic.op_induction' tactic.auto_cases_opens namespace AlgebraicGeometry +/- warning: algebraic_geometry.PresheafedSpace -> AlgebraicGeometry.PresheafedSpace is a dubious translation: +lean 3 declaration is + forall (C : Type.{u3}) [_inst_1 : CategoryTheory.Category.{u2, u3} C], Sort.{max (succ u3) (succ u2) (succ (succ u1))} +but is expected to have type + forall (C : Type.{u1}) [_inst_1 : CategoryTheory.Category.{u2, u1} C], Sort.{max (max (succ u1) (succ u2)) (succ (succ u3))} +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace AlgebraicGeometry.PresheafedSpaceₓ'. -/ /-- A `PresheafedSpace C` is a topological space equipped with a presheaf of `C`s. -/ structure PresheafedSpace where carrier : TopCat.{w} @@ -52,6 +58,12 @@ namespace PresheafedSpace attribute [protected] presheaf +/- warning: algebraic_geometry.PresheafedSpace.coe_carrier -> AlgebraicGeometry.PresheafedSpace.coeCarrier is a dubious translation: +lean 3 declaration is + forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C], Coe.{max (succ u3) (succ u2) (succ (succ u1)), succ (succ u1)} (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) TopCat.{u1} +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], CoeOut.{max (max (succ (succ u3)) (succ u2)) (succ u1), succ (succ u3)} (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) TopCat.{u3} +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.coe_carrier AlgebraicGeometry.PresheafedSpace.coeCarrierₓ'. -/ instance coeCarrier : Coe (PresheafedSpace.{w, v, u} C) TopCat.{w} where coe X := X.carrier #align algebraic_geometry.PresheafedSpace.coe_carrier AlgebraicGeometry.PresheafedSpace.coeCarrier @@ -60,6 +72,12 @@ theorem as_coe (X : PresheafedSpace.{w, v, u} C) : X.carrier = (X : TopCat.{w}) rfl #align algebraic_geometry.PresheafedSpace.as_coe AlgebraicGeometry.PresheafedSpace.as_coe +/- warning: algebraic_geometry.PresheafedSpace.mk_coe -> AlgebraicGeometry.PresheafedSpace.mk_coe is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (carrier : TopCat.{v}) (presheaf : TopCat.Presheaf.{v, v, u} C _inst_1 carrier), Eq.{succ (succ v)} TopCat.{v} ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) (AlgebraicGeometry.PresheafedSpace.mk.{v, v, u} C _inst_1 carrier presheaf)) carrier +but is expected to have type + forall {C : Type.{u_3}} [_inst_1 : CategoryTheory.Category.{u_2, u_3} C] (carrier : TopCat.{u_1}) (presheaf : TopCat.Presheaf.{u_1, u_2, u_3} C _inst_1 carrier), Eq.{succ (succ u_1)} TopCat.{u_1} (AlgebraicGeometry.PresheafedSpace.carrier.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.mk.{u_3, u_2, u_1} C _inst_1 carrier presheaf)) carrier +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.mk_coe AlgebraicGeometry.PresheafedSpace.mk_coeₓ'. -/ @[simp] theorem mk_coe (carrier) (presheaf) : (({ carrier @@ -70,6 +88,12 @@ theorem mk_coe (carrier) (presheaf) : instance (X : PresheafedSpace.{v} C) : TopologicalSpace X := X.carrier.str +/- warning: algebraic_geometry.PresheafedSpace.const -> AlgebraicGeometry.PresheafedSpace.const is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C], TopCat.{u3} -> C -> (AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1) +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], TopCat.{u3} -> C -> (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.const AlgebraicGeometry.PresheafedSpace.constₓ'. -/ /-- The constant presheaf on `X` with value `Z`. -/ def const (X : TopCat) (Z : C) : PresheafedSpace C where @@ -82,6 +106,12 @@ def const (X : TopCat) (Z : C) : PresheafedSpace C instance [Inhabited C] : Inhabited (PresheafedSpace C) := ⟨const (TopCat.of PEmpty) default⟩ +/- warning: algebraic_geometry.PresheafedSpace.hom -> AlgebraicGeometry.PresheafedSpace.Hom is a dubious translation: +lean 3 declaration is + forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C], (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) -> (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) -> Sort.{max (succ u2) (succ u1)} +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) -> (AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1) -> Sort.{max (succ u2) (succ u3)} +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.hom AlgebraicGeometry.PresheafedSpace.Homₓ'. -/ /-- A morphism between presheafed spaces `X` and `Y` consists of a continuous map `f` between the underlying topological spaces, and a (notice contravariant!) map from the presheaf on `Y` to the pushforward of the presheaf on `X` via `f`. -/ @@ -90,15 +120,27 @@ structure Hom (X Y : PresheafedSpace.{w, v, u} C) where c : Y.Presheaf ⟶ base _* X.Presheaf #align algebraic_geometry.PresheafedSpace.hom AlgebraicGeometry.PresheafedSpace.Hom +/- warning: algebraic_geometry.PresheafedSpace.ext -> AlgebraicGeometry.PresheafedSpace.Hom.ext is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {X : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} (α : AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 X Y) (β : 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Consider using '#align algebraic_geometry.PresheafedSpace.ext AlgebraicGeometry.PresheafedSpace.Hom.extₓ'. -/ @[ext] -theorem ext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base) - (h : α.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.c) : α = β := +theorem AlgebraicGeometry.PresheafedSpace.Hom.ext {X Y : PresheafedSpace C} (α β : Hom X Y) + (w : α.base = β.base) (h : α.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.c) : α = β := by cases α; cases β dsimp [presheaf.pushforward_obj] at * tidy -#align algebraic_geometry.PresheafedSpace.ext AlgebraicGeometry.PresheafedSpace.ext - +#align algebraic_geometry.PresheafedSpace.ext AlgebraicGeometry.PresheafedSpace.Hom.ext + +/- warning: algebraic_geometry.PresheafedSpace.hext -> AlgebraicGeometry.PresheafedSpace.hext is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {X : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Y : 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Consider using '#align algebraic_geometry.PresheafedSpace.hext AlgebraicGeometry.PresheafedSpace.hextₓ'. -/ -- TODO including `injections` would make tidy work earlier. theorem hext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base) (h : HEq α.c β.c) : α = β := by @@ -108,6 +150,12 @@ theorem hext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base) exacts[w, h] #align algebraic_geometry.PresheafedSpace.hext AlgebraicGeometry.PresheafedSpace.hext +/- warning: algebraic_geometry.PresheafedSpace.id -> AlgebraicGeometry.PresheafedSpace.id is a dubious translation: +lean 3 declaration is + forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (X : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1), AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 X X +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (X : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1), AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 X X +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.id AlgebraicGeometry.PresheafedSpace.idₓ'. -/ /-- The identity morphism of a `PresheafedSpace`. -/ def id (X : PresheafedSpace.{w, v, u} C) : Hom X X where @@ -115,10 +163,22 @@ def id (X : PresheafedSpace.{w, v, u} C) : Hom X X c := eqToHom (Presheaf.Pushforward.id_eq X.Presheaf).symm #align algebraic_geometry.PresheafedSpace.id AlgebraicGeometry.PresheafedSpace.id +/- warning: algebraic_geometry.PresheafedSpace.hom_inhabited -> AlgebraicGeometry.PresheafedSpace.homInhabited is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (X : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1), Inhabited.{max (succ u1) (succ u3)} (AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 X X) +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (X : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1), Inhabited.{max (succ u3) (succ u2)} (AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 X X) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.hom_inhabited AlgebraicGeometry.PresheafedSpace.homInhabitedₓ'. -/ instance homInhabited (X : PresheafedSpace C) : Inhabited (Hom X X) := ⟨id X⟩ #align algebraic_geometry.PresheafedSpace.hom_inhabited AlgebraicGeometry.PresheafedSpace.homInhabited +/- warning: algebraic_geometry.PresheafedSpace.comp -> AlgebraicGeometry.PresheafedSpace.comp is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {X : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Z : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1}, (AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 X Y) -> (AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 Y Z) -> (AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 X Z) +but is expected to have type + forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {X : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1} {Z : AlgebraicGeometry.PresheafedSpace.{u1, u2, u3} C _inst_1}, (AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 X Y) -> (AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 Y Z) -> (AlgebraicGeometry.PresheafedSpace.Hom.{u1, u2, u3} C _inst_1 X Z) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.comp AlgebraicGeometry.PresheafedSpace.compₓ'. -/ /-- Composition of morphisms of `PresheafedSpace`s. -/ def comp {X Y Z : PresheafedSpace C} (α : Hom X Y) (β : Hom Y Z) : Hom X Z where @@ -126,6 +186,12 @@ def comp {X Y Z : PresheafedSpace C} (α : Hom X Y) (β : Hom Y Z) : Hom X Z c := β.c ≫ (Presheaf.pushforward _ β.base).map α.c #align algebraic_geometry.PresheafedSpace.comp AlgebraicGeometry.PresheafedSpace.comp +/- warning: algebraic_geometry.PresheafedSpace.comp_c -> AlgebraicGeometry.PresheafedSpace.comp_c is a dubious translation: +lean 3 declaration is + forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {X : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} {Z : AlgebraicGeometry.PresheafedSpace.{u3, u1, u2} C _inst_1} (α : AlgebraicGeometry.PresheafedSpace.Hom.{u3, u1, u2} C _inst_1 X Y) (β : 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Consider using '#align algebraic_geometry.PresheafedSpace.comp_c AlgebraicGeometry.PresheafedSpace.comp_cₓ'. -/ theorem comp_c {X Y Z : PresheafedSpace C} (α : Hom X Y) (β : Hom Y Z) : (comp α β).c = β.c ≫ (Presheaf.pushforward _ β.base).map α.c := rfl @@ -137,6 +203,12 @@ section attribute [local simp] id comp +/- warning: algebraic_geometry.PresheafedSpace.category_of_PresheafedSpaces -> AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces is a dubious translation: +lean 3 declaration is + forall (C : Type.{u}) [_inst_1 : CategoryTheory.Category.{v, u} C], CategoryTheory.Category.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) +but is expected to have type + forall (C : Type.{u_1}) [_inst_1 : CategoryTheory.Category.{u_2, u_1} C], CategoryTheory.Category.{max u_2 u_3, max (max (succ u_3) u_2) u_1} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.category_of_PresheafedSpaces AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpacesₓ'. -/ /- The proofs below can be done by `tidy`, but it is too slow, and we don't have a tactic caching mechanism. -/ /-- The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map @@ -177,16 +249,34 @@ variable {C} attribute [local simp] eq_to_hom_map +/- warning: algebraic_geometry.PresheafedSpace.id_base -> AlgebraicGeometry.PresheafedSpace.id_base is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ v} (Quiver.Hom.{succ v, succ v} TopCat.{v} (CategoryTheory.CategoryStruct.toQuiver.{v, succ v} TopCat.{v} (CategoryTheory.Category.toCategoryStruct.{v, succ v} TopCat.{v} TopCat.largeCategory.{v})) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X)) (AlgebraicGeometry.PresheafedSpace.Hom.base.{v, v, u} C _inst_1 X X (CategoryTheory.CategoryStruct.id.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1)) X)) (CategoryTheory.CategoryStruct.id.{v, succ v} TopCat.{v} (CategoryTheory.Category.toCategoryStruct.{v, succ v} TopCat.{v} TopCat.largeCategory.{v}) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X)) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] (X : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1), Eq.{succ u_3} (Quiver.Hom.{succ u_3, succ u_3} TopCat.{u_3} (CategoryTheory.CategoryStruct.toQuiver.{u_3, succ u_3} TopCat.{u_3} (CategoryTheory.Category.toCategoryStruct.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3})) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.Hom.base.{u_1, u_2, u_3} C _inst_1 X X (CategoryTheory.CategoryStruct.id.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1)) X)) (CategoryTheory.CategoryStruct.id.{u_3, succ u_3} TopCat.{u_3} (CategoryTheory.Category.toCategoryStruct.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3}) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.id_base AlgebraicGeometry.PresheafedSpace.id_baseₓ'. -/ @[simp] theorem id_base (X : PresheafedSpace.{v, v, u} C) : (𝟙 X : X ⟶ X).base = 𝟙 (X : TopCat.{v}) := rfl #align algebraic_geometry.PresheafedSpace.id_base AlgebraicGeometry.PresheafedSpace.id_base +/- warning: algebraic_geometry.PresheafedSpace.id_c -> AlgebraicGeometry.PresheafedSpace.id_c is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ v} (Quiver.Hom.{succ v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X)) (CategoryTheory.CategoryStruct.toQuiver.{v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X)) (CategoryTheory.Category.toCategoryStruct.{v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 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Consider using '#align algebraic_geometry.PresheafedSpace.id_c_app AlgebraicGeometry.PresheafedSpace.id_c_appₓ'. -/ @[simp] theorem id_c_app (X : PresheafedSpace.{v, v, u} C) (U) : (𝟙 X : X ⟶ X).c.app U = @@ -204,6 +294,12 @@ theorem id_c_app (X : PresheafedSpace.{v, v, u} C) (U) : simp #align algebraic_geometry.PresheafedSpace.id_c_app AlgebraicGeometry.PresheafedSpace.id_c_app +/- warning: algebraic_geometry.PresheafedSpace.comp_base -> AlgebraicGeometry.PresheafedSpace.comp_base is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Z : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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Consider using '#align algebraic_geometry.PresheafedSpace.comp_base AlgebraicGeometry.PresheafedSpace.comp_baseₓ'. -/ @[simp] theorem comp_base {X Y Z : PresheafedSpace.{v, v, u} C} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).base = f.base ≫ g.base := @@ -217,6 +313,12 @@ theorem coe_to_fun_eq {X Y : PresheafedSpace.{v, v, u} C} (f : X ⟶ Y) : (f : X rfl #align algebraic_geometry.PresheafedSpace.coe_to_fun_eq AlgebraicGeometry.PresheafedSpace.coe_to_fun_eq +/- warning: algebraic_geometry.PresheafedSpace.comp_c_app -> AlgebraicGeometry.PresheafedSpace.comp_c_app is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Z : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (α : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max 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(AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Z)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Z))) U))))) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.comp_c_app AlgebraicGeometry.PresheafedSpace.comp_c_appₓ'. -/ -- The `reassoc` attribute was added despite the LHS not being a composition of two homs, -- for the reasons explained in the docstring. /-- Sometimes rewriting with `comp_c_app` doesn't work because of dependent type issues. @@ -228,6 +330,12 @@ theorem comp_c_app {X Y Z : PresheafedSpace.{v, v, u} C} (α : X ⟶ Y) (β : Y rfl #align algebraic_geometry.PresheafedSpace.comp_c_app AlgebraicGeometry.PresheafedSpace.comp_c_app +/- warning: algebraic_geometry.PresheafedSpace.congr_app -> AlgebraicGeometry.PresheafedSpace.congr_app is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {α : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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(TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))))))) (CategoryTheory.Functor.toPrefunctor.{u_3, u_3, u_3, u_3} (Opposite.{succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)))) (CategoryTheory.Category.opposite.{u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)))))))) (Opposite.{succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))) (CategoryTheory.Category.opposite.{u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))))) (CategoryTheory.Functor.op.{u_3, u_3, u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y))))))) (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))))))) (TopologicalSpace.Opens.map.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{u_1, u_2, u_3} C _inst_1 X Y α)))) U)) β h)))) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.congr_app AlgebraicGeometry.PresheafedSpace.congr_appₓ'. -/ theorem congr_app {X Y : PresheafedSpace.{v, v, u} C} {α β : X ⟶ Y} (h : α = β) (U) : α.c.app U = β.c.app U ≫ X.Presheaf.map (eqToHom (by subst h)) := by @@ -240,6 +348,12 @@ section variable (C) +/- warning: algebraic_geometry.PresheafedSpace.forget -> AlgebraicGeometry.PresheafedSpace.forget is a dubious translation: +lean 3 declaration is + forall (C : Type.{u}) [_inst_1 : CategoryTheory.Category.{v, u} C], CategoryTheory.Functor.{v, v, max u (succ v), succ v} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) TopCat.{v} TopCat.largeCategory.{v} +but is expected to have type + forall (C : Type.{u_1}) [_inst_1 : CategoryTheory.Category.{u_2, u_1} C], CategoryTheory.Functor.{max u_2 u_3, u_3, max (max (succ u_3) u_2) u_1, succ u_3} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) TopCat.{u_3} instTopCatLargeCategory.{u_3} +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.forget AlgebraicGeometry.PresheafedSpace.forgetₓ'. -/ /-- The forgetful functor from `PresheafedSpace` to `Top`. -/ @[simps] def forget : PresheafedSpace.{v, v, u} C ⥤ TopCat @@ -254,6 +368,12 @@ section Iso variable {X Y : PresheafedSpace.{v, v, u} C} +/- warning: algebraic_geometry.PresheafedSpace.iso_of_components -> AlgebraicGeometry.PresheafedSpace.isoOfComponents is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (H : CategoryTheory.Iso.{v, succ v} TopCat.{v} TopCat.largeCategory.{v} (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)), (CategoryTheory.Iso.{v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (TopCat.Presheaf.category.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (TopCat.Presheaf.pushforwardObj.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y) (CategoryTheory.Iso.hom.{v, succ v} TopCat.{v} TopCat.largeCategory.{v} (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y) H) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 Y)) -> (CategoryTheory.Iso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] {X : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} (H : CategoryTheory.Iso.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)), (CategoryTheory.Iso.{max u_2 u_3, max (max u_1 u_2) u_3} (TopCat.Presheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.instCategoryPresheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.Presheaf.pushforwardObj.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) (CategoryTheory.Iso.hom.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) H) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 Y)) -> (CategoryTheory.Iso.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) X Y) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.iso_of_components AlgebraicGeometry.PresheafedSpace.isoOfComponentsₓ'. -/ /-- An isomorphism of PresheafedSpaces is a homeomorphism of the underlying space, and a natural transformation between the sheaves. -/ @@ -288,6 +408,12 @@ def isoOfComponents (H : X.1 ≅ Y.1) (α : H.Hom _* X.2 ≅ Y.2) : X ≅ Y · simp #align algebraic_geometry.PresheafedSpace.iso_of_components AlgebraicGeometry.PresheafedSpace.isoOfComponents +/- warning: algebraic_geometry.PresheafedSpace.sheaf_iso_of_iso -> AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (H : CategoryTheory.Iso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y), CategoryTheory.Iso.{v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (TopCat.Presheaf.category.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 Y) (TopCat.Presheaf.pushforwardObj.{v, v, u} C _inst_1 ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{v, v, u} C _inst_1 X Y (CategoryTheory.Iso.hom.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y H)) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 X)) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] {X : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} (H : CategoryTheory.Iso.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) X Y), CategoryTheory.Iso.{max u_2 u_3, max (max u_1 u_2) u_3} (TopCat.Presheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.instCategoryPresheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 Y) (TopCat.Presheaf.pushforwardObj.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{u_1, u_2, u_3} C _inst_1 X Y (CategoryTheory.Iso.hom.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) X Y H)) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 X)) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.sheaf_iso_of_iso AlgebraicGeometry.PresheafedSpace.sheafIsoOfIsoₓ'. -/ /-- Isomorphic PresheafedSpaces have natural isomorphic presheaves. -/ @[simps] def sheafIsoOfIso (H : X ≅ Y) : Y.2 ≅ H.Hom.base _* X.2 @@ -309,14 +435,32 @@ def sheafIsoOfIso (H : X ≅ Y) : Y.2 ≅ H.Hom.base _* X.2 simpa using congr_arg (fun f => f ≫ X.presheaf.map (eq_to_hom h.symm)) this #align algebraic_geometry.PresheafedSpace.sheaf_iso_of_iso AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso +/- warning: algebraic_geometry.PresheafedSpace.base_is_iso_of_iso -> AlgebraicGeometry.PresheafedSpace.base_isIso_of_iso is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y f], CategoryTheory.IsIso.{v, succ v} TopCat.{v} TopCat.largeCategory.{v} ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{v, v, u} C _inst_1 X Y f) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] {X : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} (f : Quiver.Hom.{max (succ u_2) (succ u_3), max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) X Y f], CategoryTheory.IsIso.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{u_1, u_2, u_3} C _inst_1 X Y f) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.base_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.base_isIso_of_isoₓ'. -/ instance base_isIso_of_iso (f : X ⟶ Y) [IsIso f] : IsIso f.base := IsIso.of_iso ((forget _).mapIso (asIso f)) #align algebraic_geometry.PresheafedSpace.base_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.base_isIso_of_iso +/- warning: algebraic_geometry.PresheafedSpace.c_is_iso_of_iso -> AlgebraicGeometry.PresheafedSpace.c_isIso_of_iso is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y f], CategoryTheory.IsIso.{v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (TopCat.Presheaf.category.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y)) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 Y) (TopCat.Presheaf.pushforwardObj.{v, v, u} C _inst_1 ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{v, v, u} C _inst_1 X Y f) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.Hom.c.{v, v, u} C _inst_1 X Y f) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] {X : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1} (f : Quiver.Hom.{max (succ u_2) (succ u_3), max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1) X Y f], CategoryTheory.IsIso.{max u_2 u_3, max (max u_1 u_2) u_3} (TopCat.Presheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (TopCat.instCategoryPresheaf.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y)) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 Y) (TopCat.Presheaf.pushforwardObj.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{u_1, u_2, u_3} C _inst_1 X Y f) (AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.Hom.c.{u_1, u_2, u_3} C _inst_1 X Y f) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.c_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.c_isIso_of_isoₓ'. -/ instance c_isIso_of_iso (f : X ⟶ Y) [IsIso f] : IsIso f.c := IsIso.of_iso (sheafIsoOfIso (asIso f)) #align algebraic_geometry.PresheafedSpace.c_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.c_isIso_of_iso +/- warning: algebraic_geometry.PresheafedSpace.is_iso_of_components -> AlgebraicGeometry.PresheafedSpace.isIso_of_components is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{v, succ v} TopCat.{v} TopCat.largeCategory.{v} ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), 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((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 Y) (AlgebraicGeometry.PresheafedSpace.Hom.base.{v, v, u} C _inst_1 X Y f) (AlgebraicGeometry.PresheafedSpace.presheaf.{v, v, u} C _inst_1 X)) (AlgebraicGeometry.PresheafedSpace.Hom.c.{v, v, u} C _inst_1 X Y f)], CategoryTheory.IsIso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) X Y f +but is expected to have type + forall {C : Type.{u_3}} [_inst_1 : CategoryTheory.Category.{u_1, u_3} C] {X : AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1} (f : Quiver.Hom.{max (succ u_1) (succ u_2), max (max u_3 u_1) (succ u_2)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u_1 u_2, max (max u_3 u_1) (succ u_2)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u_1 u_2, max (max u_3 u_1) (succ u_2)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_3, u_1, u_2} C _inst_1))) X Y) [_inst_2 : CategoryTheory.IsIso.{u_2, succ u_2} TopCat.{u_2} instTopCatLargeCategory.{u_2} 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(AlgebraicGeometry.PresheafedSpace.Hom.c.{u_3, u_1, u_2} C _inst_1 X Y f)], CategoryTheory.IsIso.{max u_1 u_2, max (max u_3 u_1) (succ u_2)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_2} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_3, u_1, u_2} C _inst_1) X Y f +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.is_iso_of_components AlgebraicGeometry.PresheafedSpace.isIso_of_componentsₓ'. -/ /-- This could be used in conjunction with `category_theory.nat_iso.is_iso_of_is_iso_app`. -/ theorem isIso_of_components (f : X ⟶ Y) [IsIso f.base] [IsIso f.c] : IsIso f := by @@ -328,6 +472,12 @@ end Iso section Restrict +/- warning: algebraic_geometry.PresheafedSpace.restrict -> AlgebraicGeometry.PresheafedSpace.restrict is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {U : TopCat.{v}} (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) {f : Quiver.Hom.{succ v, succ v} TopCat.{v} (CategoryTheory.CategoryStruct.toQuiver.{v, succ v} TopCat.{v} (CategoryTheory.Category.toCategoryStruct.{v, succ v} TopCat.{v} TopCat.largeCategory.{v})) U ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => 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Consider using '#align algebraic_geometry.PresheafedSpace.of_restrict AlgebraicGeometry.PresheafedSpace.ofRestrictₓ'. -/ /-- The map from the restriction of a presheafed space. -/ @[simps] @@ -353,6 +509,12 @@ def ofRestrict {U : TopCat} (X : PresheafedSpace.{v, v, u} C) {f : U ⟶ (X : To rfl } #align algebraic_geometry.PresheafedSpace.of_restrict AlgebraicGeometry.PresheafedSpace.ofRestrict +/- warning: algebraic_geometry.PresheafedSpace.of_restrict_mono -> AlgebraicGeometry.PresheafedSpace.ofRestrict_mono is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {U : TopCat.{v}} (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (f : Quiver.Hom.{succ v, succ v} TopCat.{v} (CategoryTheory.CategoryStruct.toQuiver.{v, succ v} TopCat.{v} (CategoryTheory.Category.toCategoryStruct.{v, succ v} TopCat.{v} TopCat.largeCategory.{v})) U (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X)) (hf : OpenEmbedding.{v, 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Consider using '#align algebraic_geometry.PresheafedSpace.of_restrict_mono AlgebraicGeometry.PresheafedSpace.ofRestrict_monoₓ'. -/ instance ofRestrict_mono {U : TopCat} (X : PresheafedSpace C) (f : U ⟶ X.1) (hf : OpenEmbedding f) : Mono (X.of_restrict hf) := by @@ -386,6 +548,12 @@ instance ofRestrict_mono {U : TopCat} (X : PresheafedSpace C) (f : U ⟶ X.1) (h exact this #align algebraic_geometry.PresheafedSpace.of_restrict_mono AlgebraicGeometry.PresheafedSpace.ofRestrict_mono +/- warning: algebraic_geometry.PresheafedSpace.restrict_top_presheaf -> AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ (max u v)} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.restrict.{v, u} C _inst_1 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X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))))) (TopCat.Presheaf.pushforwardObj.{u_3, u_2, u_1} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (Prefunctor.obj.{succ u_3, succ u_3, u_3, succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CategoryTheory.CategoryStruct.toQuiver.{u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CategoryTheory.Category.toCategoryStruct.{u_3, u_3} (TopologicalSpace.Opens.{u_3} 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(CategoryTheory.Functor.toPrefunctor.{u_3, u_3, u_3, succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} 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(AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Top.top.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toTop.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))) (CategoryTheory.Iso.inv.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3} (Prefunctor.obj.{succ u_3, succ u_3, u_3, succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CategoryTheory.CategoryStruct.toQuiver.{u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CategoryTheory.Category.toCategoryStruct.{u_3, u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, 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(TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))))))))) TopCat.{u_3} (CategoryTheory.CategoryStruct.toQuiver.{u_3, succ u_3} TopCat.{u_3} (CategoryTheory.Category.toCategoryStruct.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3})) (CategoryTheory.Functor.toPrefunctor.{u_3, u_3, u_3, succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))))))) TopCat.{u_3} instTopCatLargeCategory.{u_3} (TopologicalSpace.Opens.toTopCat.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Top.top.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} 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(AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 X)) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.restrict_top_presheaf AlgebraicGeometry.PresheafedSpace.restrict_top_presheafₓ'. -/ theorem restrict_top_presheaf (X : PresheafedSpace C) : (X.restrict (Opens.openEmbedding ⊤)).Presheaf = (Opens.inclusionTopIso X.carrier).inv _* X.Presheaf := @@ -395,6 +563,12 @@ theorem restrict_top_presheaf (X : PresheafedSpace C) : rfl #align algebraic_geometry.PresheafedSpace.restrict_top_presheaf AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf +/- warning: algebraic_geometry.PresheafedSpace.of_restrict_top_c -> AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ v} (Quiver.Hom.{succ v, max u v} (TopCat.Presheaf.{v, v, u} C _inst_1 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} C _inst_1 X)) (CategoryTheory.CategoryStruct.toQuiver.{v, max u v} 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_inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))) X (TopologicalSpace.Opens.inclusion.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (Top.top.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toTop.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} 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X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))) (TopologicalSpace.Opens.openEmbedding.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (Top.top.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toTop.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C 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C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)))))) (TopologicalSpace.Opens.openEmbedding.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X) (Top.top.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toTop.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, 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_inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))))))))) TopCat.{u_3} (CategoryTheory.CategoryStruct.toQuiver.{u_3, succ u_3} TopCat.{u_3} (CategoryTheory.Category.toCategoryStruct.{u_3, succ u_3} TopCat.{u_3} instTopCatLargeCategory.{u_3})) (CategoryTheory.Functor.toPrefunctor.{u_3, u_3, u_3, succ u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (Preorder.smallCategory.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (PartialOrder.toPreorder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteSemilatticeInf.toPartialOrder.{u_3} (TopologicalSpace.Opens.{u_3} (CategoryTheory.Bundled.α.{u_3, u_3} TopologicalSpace.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))) (CompleteLattice.toCompleteSemilatticeInf.{u_3} (TopologicalSpace.Opens.{u_3} 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(AlgebraicGeometry.PresheafedSpace.presheaf.{u_1, u_2, u_3} C _inst_1 X))))))) +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.of_restrict_top_c AlgebraicGeometry.PresheafedSpace.ofRestrict_top_cₓ'. -/ theorem ofRestrict_top_c (X : PresheafedSpace C) : (X.of_restrict (Opens.openEmbedding ⊤)).c = eqToHom @@ -416,6 +590,12 @@ theorem ofRestrict_top_c (X : PresheafedSpace C) : exact ⟨fun h => ⟨⟨x, trivial⟩, h, rfl⟩, fun ⟨⟨_, _⟩, h, rfl⟩ => h⟩ #align algebraic_geometry.PresheafedSpace.of_restrict_top_c AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c +/- warning: algebraic_geometry.PresheafedSpace.to_restrict_top -> AlgebraicGeometry.PresheafedSpace.toRestrictTop is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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Consider using '#align algebraic_geometry.PresheafedSpace.to_restrict_top AlgebraicGeometry.PresheafedSpace.toRestrictTopₓ'. -/ /- or `rw [opens.inclusion_top_functor, ←comp_obj, ←opens.map_comp_eq], erw iso.inv_hom_id, cases U, refl` after `dsimp` -/ /-- The map to the restriction of a presheafed space along the canonical inclusion from the top @@ -428,6 +608,12 @@ def toRestrictTop (X : PresheafedSpace C) : X ⟶ X.restrict (Opens.openEmbeddin c := eqToHom (restrict_top_presheaf X) #align algebraic_geometry.PresheafedSpace.to_restrict_top AlgebraicGeometry.PresheafedSpace.toRestrictTop +/- warning: algebraic_geometry.PresheafedSpace.restrict_top_iso -> AlgebraicGeometry.PresheafedSpace.restrictTopIso is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), CategoryTheory.Iso.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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u_3} C _inst_1 X)) (TopCat.topologicalSpace_coe.{u_3} (AlgebraicGeometry.PresheafedSpace.carrier.{u_1, u_2, u_3} C _inst_1 X))))))) X +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.restrict_top_iso AlgebraicGeometry.PresheafedSpace.restrictTopIsoₓ'. -/ /-- The isomorphism from the restriction to the top subspace. -/ @[simps] @@ -436,14 +622,16 @@ def restrictTopIso (X : PresheafedSpace C) : X.restrict (Opens.openEmbedding ⊤ Hom := X.of_restrict _ inv := X.toRestrictTop hom_inv_id' := - ext _ _ (ConcreteCategory.hom_ext _ _ fun ⟨x, _⟩ => rfl) <| + AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ + (ConcreteCategory.hom_ext _ _ fun ⟨x, _⟩ => rfl) <| by erw [comp_c] rw [X.of_restrict_top_c] ext simp inv_hom_id' := - ext _ _ rfl <| by + AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ rfl <| + by erw [comp_c] rw [X.of_restrict_top_c] ext @@ -452,6 +640,12 @@ def restrictTopIso (X : PresheafedSpace C) : X.restrict (Opens.openEmbedding ⊤ end Restrict +/- warning: algebraic_geometry.PresheafedSpace.Γ -> AlgebraicGeometry.PresheafedSpace.Γ is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C], CategoryTheory.Functor.{v, v, max u (succ v), u} (Opposite.{succ (max u (succ v))} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1)) (CategoryTheory.Category.opposite.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1)) C _inst_1 +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C], CategoryTheory.Functor.{max u_2 u_3, u_2, max (max u_1 u_2) (succ u_3), u_1} (Opposite.{max (max (succ (succ u_3)) (succ u_2)) (succ u_1)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1)) (CategoryTheory.Category.opposite.{max u_2 u_3, max (max u_1 u_2) (succ u_3)} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_3} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_3} C _inst_1)) C _inst_1 +Case conversion may be inaccurate. Consider using '#align algebraic_geometry.PresheafedSpace.Γ AlgebraicGeometry.PresheafedSpace.Γₓ'. -/ /-- The global sections, notated Gamma. -/ @[simps] @@ -461,10 +655,22 @@ def Γ : (PresheafedSpace.{v, v, u} C)ᵒᵖ ⥤ C map X Y f := f.unop.c.app (op ⊤) #align algebraic_geometry.PresheafedSpace.Γ AlgebraicGeometry.PresheafedSpace.Γ +/- warning: algebraic_geometry.PresheafedSpace.Γ_obj_op -> AlgebraicGeometry.PresheafedSpace.Γ_obj_op is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ u} C (CategoryTheory.Functor.obj.{v, v, max u (succ v), u} (Opposite.{succ (max u (succ v))} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1)) (CategoryTheory.Category.opposite.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1)) C _inst_1 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Consider using '#align algebraic_geometry.PresheafedSpace.Γ_obj_op AlgebraicGeometry.PresheafedSpace.Γ_obj_opₓ'. -/ theorem Γ_obj_op (X : PresheafedSpace C) : Γ.obj (op X) = X.Presheaf.obj (op ⊤) := rfl #align algebraic_geometry.PresheafedSpace.Γ_obj_op AlgebraicGeometry.PresheafedSpace.Γ_obj_op +/- warning: algebraic_geometry.PresheafedSpace.Γ_map_op -> AlgebraicGeometry.PresheafedSpace.Γ_map_op is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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Consider using '#align algebraic_geometry.PresheafedSpace.Γ_map_op AlgebraicGeometry.PresheafedSpace.Γ_map_opₓ'. -/ theorem Γ_map_op {X Y : PresheafedSpace.{v, v, u} C} (f : X ⟶ Y) : Γ.map f.op = f.c.app (op ⊤) := rfl #align algebraic_geometry.PresheafedSpace.Γ_map_op AlgebraicGeometry.PresheafedSpace.Γ_map_op @@ -485,6 +691,12 @@ attribute [local simp] presheaf.pushforward_obj namespace Functor +/- warning: category_theory.functor.map_presheaf -> CategoryTheory.Functor.mapPresheaf is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D], (CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) -> (CategoryTheory.Functor.{v, v, max u (succ v), max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} D _inst_2)) +but is expected to have type + forall {C : Type.{u_1}} [_inst_1 : CategoryTheory.Category.{u_2, u_1} C] {D : Type.{u_3}} [_inst_2 : CategoryTheory.Category.{u_4, u_3} D], (CategoryTheory.Functor.{u_2, u_4, u_1, u_3} C _inst_1 D _inst_2) -> (CategoryTheory.Functor.{max u_2 u_5, max u_4 u_5, max (max (succ u_5) u_2) u_1, max (max (succ u_5) u_4) u_3} (AlgebraicGeometry.PresheafedSpace.{u_1, u_2, u_5} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_1, u_2, u_5} C _inst_1) (AlgebraicGeometry.PresheafedSpace.{u_3, u_4, u_5} D _inst_2) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_3, u_4, u_5} D _inst_2)) +Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_presheaf CategoryTheory.Functor.mapPresheafₓ'. -/ /-- We can apply a functor `F : C ⥤ D` to the values of the presheaf in any `PresheafedSpace C`, giving a functor `PresheafedSpace C ⥤ PresheafedSpace D` -/ def mapPresheaf (F : C ⥤ D) : PresheafedSpace.{v, v, u} C ⥤ PresheafedSpace.{v, v, u} D @@ -497,24 +709,48 @@ def mapPresheaf (F : C ⥤ D) : PresheafedSpace.{v, v, u} C ⥤ PresheafedSpace. c := whiskerRight f.c F } #align category_theory.functor.map_presheaf CategoryTheory.Functor.mapPresheaf +/- warning: category_theory.functor.map_presheaf_obj_X -> CategoryTheory.Functor.mapPresheaf_obj_X is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D] (F : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ (succ v)} TopCat.{v} ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} D _inst_2)))) (CategoryTheory.Functor.obj.{v, v, max u (succ v), max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} C _inst_1) (AlgebraicGeometry.PresheafedSpace.{v, v, u} D _inst_2) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{v, u} D _inst_2) (CategoryTheory.Functor.mapPresheaf.{v, u} C _inst_1 D _inst_2 F) X)) ((fun (a : Sort.{max (succ u) (succ (succ v))}) (b : Type.{succ v}) [self : HasLiftT.{max (succ u) (succ (succ v)), succ (succ v)} a b] => self.0) (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (HasLiftT.mk.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (CoeTCₓ.coe.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (coeBase.{max (succ u) (succ (succ v)), succ (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) TopCat.{v} (AlgebraicGeometry.PresheafedSpace.coeCarrier.{v, v, u} C _inst_1)))) X) +but is expected to have type + forall {C : Type.{u_3}} [_inst_1 : CategoryTheory.Category.{u_1, u_3} C] {D : Type.{u_4}} [_inst_2 : CategoryTheory.Category.{u_2, u_4} D] (F : CategoryTheory.Functor.{u_1, u_2, u_3, u_4} C _inst_1 D _inst_2) (X : AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_5} C _inst_1), Eq.{succ (succ u_5)} TopCat.{u_5} (AlgebraicGeometry.PresheafedSpace.carrier.{u_4, u_2, u_5} D _inst_2 (Prefunctor.obj.{max (succ u_1) (succ u_5), max (succ u_2) (succ u_5), max (max u_3 u_1) (succ u_5), max (max u_4 u_2) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_5} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u_1 u_5, max (max u_3 u_1) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_5} C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u_1 u_5, max (max u_3 u_1) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_5} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_3, u_1, u_5} C _inst_1))) (AlgebraicGeometry.PresheafedSpace.{u_4, u_2, u_5} D _inst_2) (CategoryTheory.CategoryStruct.toQuiver.{max u_2 u_5, max (max u_4 u_2) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_4, u_2, u_5} D _inst_2) (CategoryTheory.Category.toCategoryStruct.{max u_2 u_5, max (max u_4 u_2) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_4, u_2, u_5} D _inst_2) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_4, u_2, u_5} D _inst_2))) (CategoryTheory.Functor.toPrefunctor.{max u_1 u_5, max u_2 u_5, max (max u_3 u_1) (succ u_5), max (max u_4 u_2) (succ u_5)} (AlgebraicGeometry.PresheafedSpace.{u_3, u_1, u_5} C _inst_1) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_3, u_1, u_5} C _inst_1) (AlgebraicGeometry.PresheafedSpace.{u_4, u_2, u_5} D _inst_2) (AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces.{u_4, u_2, u_5} D _inst_2) (CategoryTheory.Functor.mapPresheaf.{u_3, u_1, u_4, u_2, u_5} C _inst_1 D _inst_2 F)) X)) (AlgebraicGeometry.PresheafedSpace.carrier.{u_3, u_1, u_5} C _inst_1 X) +Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_presheaf_obj_X CategoryTheory.Functor.mapPresheaf_obj_Xₓ'. -/ @[simp] theorem mapPresheaf_obj_X (F : C ⥤ D) (X : PresheafedSpace C) : (F.mapPresheaf.obj X : TopCat.{v}) = (X : TopCat.{v}) := rfl #align category_theory.functor.map_presheaf_obj_X CategoryTheory.Functor.mapPresheaf_obj_X +/- warning: category_theory.functor.map_presheaf_obj_presheaf -> CategoryTheory.Functor.mapPresheaf_obj_presheaf is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D] (F : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) (X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1), Eq.{succ (max u v)} (TopCat.Presheaf.{v, v, u} D _inst_2 (AlgebraicGeometry.PresheafedSpace.carrier.{v, v, u} D _inst_2 (CategoryTheory.Functor.obj.{v, v, max u (succ v), max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) 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Consider using '#align category_theory.functor.map_presheaf_obj_presheaf CategoryTheory.Functor.mapPresheaf_obj_presheafₓ'. -/ @[simp] theorem mapPresheaf_obj_presheaf (F : C ⥤ D) (X : PresheafedSpace C) : (F.mapPresheaf.obj X).Presheaf = X.Presheaf ⋙ F := rfl #align category_theory.functor.map_presheaf_obj_presheaf CategoryTheory.Functor.mapPresheaf_obj_presheaf +/- warning: category_theory.functor.map_presheaf_map_f -> CategoryTheory.Functor.mapPresheaf_map_f is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D] (F : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} 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Consider using '#align category_theory.functor.map_presheaf_map_f CategoryTheory.Functor.mapPresheaf_map_fₓ'. -/ @[simp] theorem mapPresheaf_map_f (F : C ⥤ D) {X Y : PresheafedSpace.{v, v, u} C} (f : X ⟶ Y) : (F.mapPresheaf.map f).base = f.base := rfl #align category_theory.functor.map_presheaf_map_f CategoryTheory.Functor.mapPresheaf_map_f +/- warning: category_theory.functor.map_presheaf_map_c -> CategoryTheory.Functor.mapPresheaf_map_c is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D] (F : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) {X : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} {Y : AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1} (f : Quiver.Hom.{succ v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, v, u} C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (AlgebraicGeometry.PresheafedSpace.{v, 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Consider using '#align category_theory.functor.map_presheaf_map_c CategoryTheory.Functor.mapPresheaf_map_cₓ'. -/ @[simp] theorem mapPresheaf_map_c (F : C ⥤ D) {X Y : PresheafedSpace.{v, v, u} C} (f : X ⟶ Y) : (F.mapPresheaf.map f).c = whiskerRight f.c F := @@ -525,6 +761,12 @@ end Functor namespace NatTrans +/- warning: category_theory.nat_trans.on_presheaf -> CategoryTheory.NatTrans.onPresheaf is a dubious translation: +lean 3 declaration is + forall {C : Type.{u}} [_inst_1 : CategoryTheory.Category.{v, u} C] {D : Type.{u}} [_inst_2 : CategoryTheory.Category.{v, u} D] {F : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2} {G : CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2}, (Quiver.Hom.{succ (max u v), max v u} (CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) (CategoryTheory.CategoryStruct.toQuiver.{max u v, max v u} (CategoryTheory.Functor.{v, v, u, u} C _inst_1 D _inst_2) (CategoryTheory.Category.toCategoryStruct.{max u v, max v u} 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Consider using '#align category_theory.nat_trans.on_presheaf CategoryTheory.NatTrans.onPresheafₓ'. -/ /-- A natural transformation induces a natural transformation between the `map_presheaf` functors. -/ def onPresheaf {F G : C ⥤ D} (α : F ⟶ G) : G.mapPresheaf ⟶ F.mapPresheaf diff --git a/Mathbin/AlgebraicGeometry/PresheafedSpace/Gluing.lean b/Mathbin/AlgebraicGeometry/PresheafedSpace/Gluing.lean index dfc3c8b903..bc5e4e65a8 100644 --- a/Mathbin/AlgebraicGeometry/PresheafedSpace/Gluing.lean +++ b/Mathbin/AlgebraicGeometry/PresheafedSpace/Gluing.lean @@ -575,11 +575,11 @@ theorem ι_isoPresheafedSpace_inv (i : D.J) : 𝖣.ι_gluedIso_inv _ _ #align algebraic_geometry.SheafedSpace.glue_data.ι_iso_PresheafedSpace_inv AlgebraicGeometry.SheafedSpace.GlueData.ι_isoPresheafedSpace_inv -instance ι_isOpenImmersion (i : D.J) : IsOpenImmersion (𝖣.ι i) := +instance ιIsOpenImmersion (i : D.J) : IsOpenImmersion (𝖣.ι i) := by rw [← D.ι_iso_PresheafedSpace_inv] infer_instance -#align algebraic_geometry.SheafedSpace.glue_data.ι_is_open_immersion AlgebraicGeometry.SheafedSpace.GlueData.ι_isOpenImmersion +#align algebraic_geometry.SheafedSpace.glue_data.ι_is_open_immersion AlgebraicGeometry.SheafedSpace.GlueData.ιIsOpenImmersion theorem ι_jointly_surjective (x : 𝖣.glued) : ∃ (i : D.J)(y : D.U i), (𝖣.ι i).base y = x := 𝖣.ι_jointly_surjective (SheafedSpace.forget _ ⋙ CategoryTheory.forget TopCat) x diff --git a/Mathbin/AlgebraicGeometry/Spec.lean b/Mathbin/AlgebraicGeometry/Spec.lean index 8b690a43aa..1713c10665 100644 --- a/Mathbin/AlgebraicGeometry/Spec.lean +++ b/Mathbin/AlgebraicGeometry/Spec.lean @@ -111,7 +111,7 @@ def Spec.sheafedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) : @[simp] theorem Spec.sheafedSpaceMap_id {R : CommRingCat} : Spec.sheafedSpaceMap (𝟙 R) = 𝟙 (Spec.sheafedSpaceObj R) := - PresheafedSpace.ext _ _ (Spec.topMap_id R) <| + AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ (Spec.topMap_id R) <| NatTrans.ext _ _ <| funext fun U => by dsimp @@ -122,7 +122,7 @@ theorem Spec.sheafedSpaceMap_id {R : CommRingCat} : theorem Spec.sheafedSpaceMap_comp {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T) : Spec.sheafedSpaceMap (f ≫ g) = Spec.sheafedSpaceMap g ≫ Spec.sheafedSpaceMap f := - PresheafedSpace.ext _ _ (Spec.topMap_comp f g) <| + AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ (Spec.topMap_comp f g) <| NatTrans.ext _ _ <| funext fun U => by dsimp diff --git a/Mathbin/Analysis/LocallyConvex/WeakDual.lean b/Mathbin/Analysis/LocallyConvex/WeakDual.lean index 01155ff634..dfd3f4957e 100644 --- a/Mathbin/Analysis/LocallyConvex/WeakDual.lean +++ b/Mathbin/Analysis/LocallyConvex/WeakDual.lean @@ -159,7 +159,7 @@ variable [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [M variable [Nonempty ι] [NormedSpace ℝ 𝕜] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] instance {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} : LocallyConvexSpace ℝ (WeakBilin B) := - B.weakBilin_withSeminorms.toLocallyConvexSpace + B.weakBilin_withSeminorms.to_locallyConvexSpace end LocallyConvex diff --git a/Mathbin/Analysis/LocallyConvex/WithSeminorms.lean b/Mathbin/Analysis/LocallyConvex/WithSeminorms.lean index 1a8c50d898..2c02ccbebc 100644 --- a/Mathbin/Analysis/LocallyConvex/WithSeminorms.lean +++ b/Mathbin/Analysis/LocallyConvex/WithSeminorms.lean @@ -683,7 +683,7 @@ open LocallyConvexSpace variable [Nonempty ι] [NormedField 𝕜] [NormedSpace ℝ 𝕜] [AddCommGroup E] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] -theorem WithSeminorms.toLocallyConvexSpace {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) : +theorem WithSeminorms.to_locallyConvexSpace {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) : LocallyConvexSpace ℝ E := by apply of_basis_zero ℝ E id fun s => s ∈ p.basis_sets @@ -694,7 +694,7 @@ theorem WithSeminorms.toLocallyConvexSpace {p : SeminormFamily 𝕜 E ι} (hp : simp_rw [Set.mem_iUnion, Set.mem_singleton_iff] at hs rcases hs with ⟨I, r, hr, rfl⟩ exact convex_ball _ _ _ -#align with_seminorms.to_locally_convex_space WithSeminorms.toLocallyConvexSpace +#align with_seminorms.to_locally_convex_space WithSeminorms.to_locallyConvexSpace end LocallyConvexSpace @@ -704,16 +704,16 @@ variable (𝕜) [NormedField 𝕜] [NormedSpace ℝ 𝕜] [SeminormedAddCommGrou /-- Not an instance since `𝕜` can't be inferred. See `normed_space.to_locally_convex_space` for a slightly weaker instance version. -/ -theorem NormedSpace.toLocallyConvexSpace' [NormedSpace 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] : +theorem NormedSpace.to_locally_convex_space' [NormedSpace 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] : LocallyConvexSpace ℝ E := - (norm_withSeminorms 𝕜 E).toLocallyConvexSpace -#align normed_space.to_locally_convex_space' NormedSpace.toLocallyConvexSpace' + (norm_withSeminorms 𝕜 E).to_locallyConvexSpace +#align normed_space.to_locally_convex_space' NormedSpace.to_locally_convex_space' /-- See `normed_space.to_locally_convex_space'` for a slightly stronger version which is not an instance. -/ -instance NormedSpace.toLocallyConvexSpace [NormedSpace ℝ E] : LocallyConvexSpace ℝ E := - NormedSpace.toLocallyConvexSpace' ℝ -#align normed_space.to_locally_convex_space NormedSpace.toLocallyConvexSpace +instance NormedSpace.to_locallyConvexSpace [NormedSpace ℝ E] : LocallyConvexSpace ℝ E := + NormedSpace.to_locally_convex_space' ℝ +#align normed_space.to_locally_convex_space NormedSpace.to_locallyConvexSpace end NormedSpace diff --git a/Mathbin/Analysis/SchwartzSpace.lean b/Mathbin/Analysis/SchwartzSpace.lean index cc49a2f740..e24ac8aace 100644 --- a/Mathbin/Analysis/SchwartzSpace.lean +++ b/Mathbin/Analysis/SchwartzSpace.lean @@ -597,7 +597,7 @@ instance : UniformAddGroup 𝓢(E, F) := (schwartzSeminormFamily ℝ E F).AddGroupFilterBasis.UniformAddGroup instance : LocallyConvexSpace ℝ 𝓢(E, F) := - (schwartz_withSeminorms ℝ E F).toLocallyConvexSpace + (schwartz_withSeminorms ℝ E F).to_locallyConvexSpace instance : TopologicalSpace.FirstCountableTopology 𝓢(E, F) := (schwartz_withSeminorms ℝ E F).first_countable diff --git a/Mathbin/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean b/Mathbin/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean index 814ca789a5..547245f7e1 100644 --- a/Mathbin/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean +++ b/Mathbin/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean @@ -29,11 +29,23 @@ variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpa [OpensMeasurableSpace E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [MeasurableSpace F] [BorelSpace F] +/- warning: continuous_linear_map.measurable -> ContinuousLinearMap.measurable is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} [_inst_2 : NormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] [_inst_5 : MeasurableSpace.{u2} E] [_inst_6 : OpensMeasurableSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) _inst_5] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u1, u3} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] [_inst_9 : MeasurableSpace.{u3} F] [_inst_10 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) _inst_9] (L : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E 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_inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u1, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8)) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) (NormedSpace.toModule.{u1, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8)) L) +but is expected to have type + forall {𝕜 : Type.{u3}} [_inst_2 : NormedField.{u3} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] [_inst_5 : MeasurableSpace.{u2} E] [_inst_6 : OpensMeasurableSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) _inst_5] {F : Type.{u1}} [_inst_7 : NormedAddCommGroup.{u1} F] [_inst_8 : NormedSpace.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)] [_inst_9 : MeasurableSpace.{u1} F] [_inst_10 : BorelSpace.{u1} F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) _inst_9] (L : ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8)), Measurable.{u2, u1} E F _inst_5 _inst_9 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8)) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8)) E F (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))) (RingHom.id.{u3} 𝕜 (Semiring.toNonAssocSemiring.{u3} 𝕜 (DivisionSemiring.toSemiring.{u3} 𝕜 (Semifield.toDivisionSemiring.{u3} 𝕜 (Field.toSemifield.{u3} 𝕜 (NormedField.toField.{u3} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u1} F (PseudoMetricSpace.toUniformSpace.{u1} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u1} F (NormedAddCommGroup.toAddCommGroup.{u1} F _inst_7)) (NormedSpace.toModule.{u3, u2} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u3, u1} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} F _inst_7) _inst_8)))) L) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.measurable ContinuousLinearMap.measurableₓ'. -/ @[measurability] protected theorem measurable (L : E →L[𝕜] F) : Measurable L := L.Continuous.Measurable #align continuous_linear_map.measurable ContinuousLinearMap.measurable +/- warning: continuous_linear_map.measurable_comp -> ContinuousLinearMap.measurable_comp is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u2}} [_inst_2 : NormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_3 : NormedAddCommGroup.{u3} E] [_inst_4 : NormedSpace.{u2, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)] [_inst_5 : MeasurableSpace.{u3} E] [_inst_6 : OpensMeasurableSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) _inst_5] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] [_inst_9 : MeasurableSpace.{u4} F] [_inst_10 : BorelSpace.{u4} F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) _inst_9] (L : ContinuousLinearMap.{u2, u2, u3, u4} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u2, u4} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8)) {φ : α -> E}, (Measurable.{u1, u3} α E _inst_1 _inst_5 φ) -> (Measurable.{u1, u4} α F _inst_1 _inst_9 (fun (a : α) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u2, u2, u3, u4} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u2, u4} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8)) (fun (_x : ContinuousLinearMap.{u2, u2, u3, u4} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u2, u4} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8)) => E -> F) (ContinuousLinearMap.toFun.{u2, u2, u3, u4} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) (NormedSpace.toModule.{u2, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u2, u4} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8)) L (φ a))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u4}} [_inst_2 : NormedField.{u4} 𝕜] {E : Type.{u3}} [_inst_3 : NormedAddCommGroup.{u3} E] [_inst_4 : NormedSpace.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)] [_inst_5 : MeasurableSpace.{u3} E] [_inst_6 : OpensMeasurableSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) _inst_5] {F : Type.{u2}} [_inst_7 : NormedAddCommGroup.{u2} F] [_inst_8 : NormedSpace.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)] [_inst_9 : MeasurableSpace.{u2} F] [_inst_10 : BorelSpace.{u2} F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) _inst_9] (L : ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8)) {φ : α -> E}, (Measurable.{u1, u3} α E _inst_1 _inst_5 φ) -> (Measurable.{u1, u2} α F _inst_1 _inst_9 (fun (a : α) => FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8)) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8)) E F (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 _inst_2)))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_7)) (NormedSpace.toModule.{u4, u3} 𝕜 E _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u4, u2} 𝕜 F _inst_2 (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_7) _inst_8)))) L (φ a))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.measurable_comp ContinuousLinearMap.measurable_compₓ'. -/ theorem measurable_comp (L : E →L[𝕜] F) {φ : α → E} (φ_meas : Measurable φ) : Measurable fun a : α => L (φ a) := L.Measurable.comp φ_meas @@ -54,18 +66,36 @@ instance : MeasurableSpace (E →L[𝕜] F) := instance : BorelSpace (E →L[𝕜] F) := ⟨rfl⟩ +/- warning: continuous_linear_map.measurable_apply -> ContinuousLinearMap.measurable_apply is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] {F : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} F] [_inst_6 : NormedSpace.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)] [_inst_7 : MeasurableSpace.{u3} F] [_inst_8 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) _inst_7] (x : E), Measurable.{max u2 u3, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) F (ContinuousLinearMap.instMeasurableSpace.{u1, u2, u3} 𝕜 _inst_2 E _inst_3 _inst_4 F _inst_5 _inst_6) _inst_7 (fun (f : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) f x) +but is expected to have type + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] {F : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} F] [_inst_6 : NormedSpace.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)] [_inst_7 : MeasurableSpace.{u3} F] [_inst_8 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) _inst_7] (x : E), Measurable.{max u2 u3, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) x) (ContinuousLinearMap.instMeasurableSpace.{u1, u2, u3} 𝕜 _inst_2 E _inst_3 _inst_4 F _inst_5 _inst_6) _inst_7 (fun (f : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) E F (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u1, u1, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)))) f x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.measurable_apply ContinuousLinearMap.measurable_applyₓ'. -/ @[measurability] theorem measurable_apply [MeasurableSpace F] [BorelSpace F] (x : E) : Measurable fun f : E →L[𝕜] F => f x := (apply 𝕜 F x).Continuous.Measurable #align continuous_linear_map.measurable_apply ContinuousLinearMap.measurable_apply +/- warning: continuous_linear_map.measurable_apply' -> ContinuousLinearMap.measurable_apply' is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] {F : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} F] [_inst_6 : NormedSpace.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)] [_inst_7 : MeasurableSpace.{u2} E] [_inst_8 : OpensMeasurableSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) _inst_7] [_inst_9 : MeasurableSpace.{u3} F] [_inst_10 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) _inst_9], Measurable.{u2, max u2 u3} E ((ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) -> F) _inst_7 (MeasurableSpace.pi.{max u2 u3, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (fun (f : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => F) (fun (a : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => _inst_9)) (fun (x : E) (f : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) f x) +but is expected to have type + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u3}} [_inst_3 : NormedAddCommGroup.{u3} E] [_inst_4 : NormedSpace.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)] {F : Type.{u2}} [_inst_5 : NormedAddCommGroup.{u2} F] [_inst_6 : NormedSpace.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)] [_inst_7 : MeasurableSpace.{u3} E] [_inst_8 : OpensMeasurableSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) _inst_7] [_inst_9 : MeasurableSpace.{u2} F] [_inst_10 : BorelSpace.{u2} F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) _inst_9], Measurable.{u3, max u3 u2} E ((ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) -> F) _inst_7 (MeasurableSpace.pi.{max u3 u2, u2} (ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) (fun (f : ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) => F) (fun (a : ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) => _inst_9)) (fun (x : E) (f : ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) => FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E 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_inst_3)))) (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u1, u1, u3, u2} (ContinuousLinearMap.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E 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(NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u2} F (PseudoMetricSpace.toUniformSpace.{u2} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u2} F (NormedAddCommGroup.toAddCommGroup.{u2} F _inst_5)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u2} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} F _inst_5) _inst_6)))) f x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.measurable_apply' ContinuousLinearMap.measurable_apply'ₓ'. -/ @[measurability] theorem measurable_apply' [MeasurableSpace E] [OpensMeasurableSpace E] [MeasurableSpace F] [BorelSpace F] : Measurable fun (x : E) (f : E →L[𝕜] F) => f x := measurable_pi_lambda _ fun f => f.Measurable #align continuous_linear_map.measurable_apply' ContinuousLinearMap.measurable_apply' +/- warning: continuous_linear_map.measurable_coe -> ContinuousLinearMap.measurable_coe is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] {F : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} F] [_inst_6 : NormedSpace.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)] [_inst_7 : MeasurableSpace.{u3} F] [_inst_8 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) _inst_7], Measurable.{max u2 u3, max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (E -> F) (ContinuousLinearMap.instMeasurableSpace.{u1, u2, u3} 𝕜 _inst_2 E _inst_3 _inst_4 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(NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) f x) +but is expected to have type + forall {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] {F : Type.{u3}} [_inst_5 : NormedAddCommGroup.{u3} F] [_inst_6 : NormedSpace.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)] [_inst_7 : MeasurableSpace.{u3} F] [_inst_8 : BorelSpace.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) _inst_7], Measurable.{max u2 u3, max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (forall (x : E), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) x) (ContinuousLinearMap.instMeasurableSpace.{u1, u2, u3} 𝕜 _inst_2 E _inst_3 _inst_4 F _inst_5 _inst_6) (MeasurableSpace.pi.{u2, u3} E (fun (x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) x) (fun (a : E) => _inst_7)) (fun (f : ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) (x : E) => FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) E F (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u1, u1, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u3} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_5)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (NormedSpace.toModule.{u1, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_5) _inst_6)))) f x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.measurable_coe ContinuousLinearMap.measurable_coeₓ'. -/ @[measurability] theorem measurable_coe [MeasurableSpace F] [BorelSpace F] : Measurable fun (f : E →L[𝕜] F) (x : E) => f x := @@ -81,12 +111,24 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] +/- warning: measurable.apply_continuous_linear_map -> Measurable.apply_continuousLinearMap is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u2}} [_inst_2 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_3 : NormedAddCommGroup.{u3} E] [_inst_4 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)] [_inst_5 : MeasurableSpace.{u3} E] [_inst_6 : BorelSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) _inst_5] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4))}, (Measurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) _inst_1 (ContinuousLinearMap.instMeasurableSpace.{u2, u4, u3} 𝕜 _inst_2 F _inst_7 _inst_8 E _inst_3 _inst_4) φ) -> (forall (v : F), Measurable.{u1, u3} α E _inst_1 _inst_5 (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) (φ a) v)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u4}} [_inst_2 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] [_inst_5 : MeasurableSpace.{u2} E] [_inst_6 : BorelSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) _inst_5] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4))}, (Measurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) _inst_1 (ContinuousLinearMap.instMeasurableSpace.{u4, u3, u2} 𝕜 _inst_2 F _inst_7 _inst_8 E _inst_3 _inst_4) φ) -> (forall (v : F), Measurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_1 _inst_5 (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)))) (φ a) v)) +Case conversion may be inaccurate. Consider using '#align measurable.apply_continuous_linear_map Measurable.apply_continuousLinearMapₓ'. -/ @[measurability] theorem Measurable.apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : Measurable φ) (v : F) : Measurable fun a => φ a v := (ContinuousLinearMap.apply 𝕜 E v).Measurable.comp hφ #align measurable.apply_continuous_linear_map Measurable.apply_continuousLinearMap +/- warning: ae_measurable.apply_continuous_linear_map -> AEMeasurable.apply_continuousLinearMap is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u2}} [_inst_2 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u3}} [_inst_3 : NormedAddCommGroup.{u3} E] [_inst_4 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)] [_inst_5 : MeasurableSpace.{u3} E] [_inst_6 : BorelSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) _inst_5] {F : Type.{u4}} [_inst_7 : NormedAddCommGroup.{u4} F] [_inst_8 : NormedSpace.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4))} {μ : MeasureTheory.Measure.{u1} α _inst_1}, (AEMeasurable.{u1, max u4 u3} α (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) (ContinuousLinearMap.instMeasurableSpace.{u2, u4, u3} 𝕜 _inst_2 F _inst_7 _inst_8 E _inst_3 _inst_4) _inst_1 φ μ) -> (forall (v : F), AEMeasurable.{u1, u3} α E _inst_5 _inst_1 (fun (a : α) => coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) (fun (_x : ContinuousLinearMap.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) => F -> E) (ContinuousLinearMap.toFun.{u2, u2, u4, u3} 𝕜 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_3)) (NormedSpace.toModule.{u2, u4} 𝕜 F (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_7) _inst_8) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_3) _inst_4)) (φ a) v) μ) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u4}} [_inst_2 : NontriviallyNormedField.{u4} 𝕜] {E : Type.{u2}} [_inst_3 : NormedAddCommGroup.{u2} E] [_inst_4 : NormedSpace.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)] [_inst_5 : MeasurableSpace.{u2} E] [_inst_6 : BorelSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) _inst_5] {F : Type.{u3}} [_inst_7 : NormedAddCommGroup.{u3} F] [_inst_8 : NormedSpace.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)] {φ : α -> (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4))} {μ : MeasureTheory.Measure.{u1} α _inst_1}, (AEMeasurable.{u1, max u2 u3} α (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) (ContinuousLinearMap.instMeasurableSpace.{u4, u3, u2} 𝕜 _inst_2 F _inst_7 _inst_8 E _inst_3 _inst_4) _inst_1 φ μ) -> (forall (v : F), AEMeasurable.{u1, u2} α ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) v) _inst_5 _inst_1 (fun (a : α) => FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => E) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) F E (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u3, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))) (RingHom.id.{u4} 𝕜 (Semiring.toNonAssocSemiring.{u4} 𝕜 (DivisionSemiring.toSemiring.{u4} 𝕜 (Semifield.toDivisionSemiring.{u4} 𝕜 (Field.toSemifield.{u4} 𝕜 (NormedField.toField.{u4} 𝕜 (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_7)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_3)) (NormedSpace.toModule.{u4, u3} 𝕜 F (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_7) _inst_8) (NormedSpace.toModule.{u4, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u4} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_3) _inst_4)))) (φ a) v) μ) +Case conversion may be inaccurate. Consider using '#align ae_measurable.apply_continuous_linear_map AEMeasurable.apply_continuousLinearMapₓ'. -/ @[measurability] theorem AEMeasurable.apply_continuousLinearMap {φ : α → F →L[𝕜] E} {μ : Measure α} (hφ : AEMeasurable φ μ) (v : F) : AEMeasurable (fun a => φ a v) μ := @@ -102,15 +144,27 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [Me variable [BorelSpace 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] +/- warning: measurable_smul_const -> measurable_smul_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u2}} [_inst_2 : NontriviallyNormedField.{u2} 𝕜] [_inst_3 : CompleteSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))] [_inst_4 : MeasurableSpace.{u2} 𝕜] [_inst_5 : BorelSpace.{u2} 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) _inst_4] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] [_inst_8 : MeasurableSpace.{u3} E] [_inst_9 : BorelSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) _inst_8] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (OfNat.mk.{u3} E 0 (Zero.zero.{u3} E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (SubNegMonoid.toAddMonoid.{u3} E (AddGroup.toSubNegMonoid.{u3} E (NormedAddGroup.toAddGroup.{u3} E (NormedAddCommGroup.toNormedAddGroup.{u3} E _inst_6)))))))))) -> (Iff (Measurable.{u1, u3} α E _inst_1 _inst_8 (fun (x : α) => SMul.smul.{u2, u3} 𝕜 E (SMulZeroClass.toHasSmul.{u2, u3} 𝕜 E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} 𝕜 E (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2)))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u2, u3} 𝕜 E (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7))))) (f x) c)) (Measurable.{u1, u2} α 𝕜 _inst_1 _inst_4 f)) +but is expected to have type + forall {α : Type.{u2}} [_inst_1 : MeasurableSpace.{u2} α] {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] [_inst_3 : CompleteSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))] [_inst_4 : MeasurableSpace.{u1} 𝕜] [_inst_5 : BorelSpace.{u1} 𝕜 (UniformSpace.toTopologicalSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) _inst_4] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] [_inst_8 : MeasurableSpace.{u3} E] [_inst_9 : BorelSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) _inst_8] {f : α -> 𝕜} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (Zero.toOfNat0.{u3} E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6))))))))) -> (Iff (Measurable.{u2, u3} α E _inst_1 _inst_8 (fun (x : α) => HSMul.hSMul.{u1, u3, u3} 𝕜 E E (instHSMul.{u1, u3} 𝕜 E (SMulZeroClass.toSMul.{u1, u3} 𝕜 E (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSMulZeroClass.{u1, u3} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2)))))) (NegZeroClass.toZero.{u3} E (SubNegZeroMonoid.toNegZeroClass.{u3} E (SubtractionMonoid.toSubNegZeroMonoid.{u3} E (SubtractionCommMonoid.toSubtractionMonoid.{u3} E (AddCommGroup.toDivisionAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u1, u3} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (AddCommGroup.toAddCommMonoid.{u3} E (NormedAddCommGroup.toAddCommGroup.{u3} E _inst_6)) (NormedSpace.toModule.{u1, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7)))))) (f x) c)) (Measurable.{u2, u1} α 𝕜 _inst_1 _inst_4 f)) +Case conversion may be inaccurate. Consider using '#align measurable_smul_const measurable_smul_constₓ'. -/ theorem measurable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) : (Measurable fun x => f x • c) ↔ Measurable f := (closedEmbedding_smul_left hc).MeasurableEmbedding.measurable_comp_iff #align measurable_smul_const measurable_smul_const -theorem aEMeasurable_smul_const {f : α → 𝕜} {μ : Measure α} {c : E} (hc : c ≠ 0) : +/- warning: ae_measurable_smul_const -> aemeasurable_smul_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {𝕜 : Type.{u2}} [_inst_2 : NontriviallyNormedField.{u2} 𝕜] [_inst_3 : CompleteSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))] [_inst_4 : MeasurableSpace.{u2} 𝕜] [_inst_5 : BorelSpace.{u2} 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSemiNormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))) _inst_4] {E : Type.{u3}} [_inst_6 : NormedAddCommGroup.{u3} E] [_inst_7 : NormedSpace.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)] [_inst_8 : MeasurableSpace.{u3} E] [_inst_9 : BorelSpace.{u3} E (UniformSpace.toTopologicalSpace.{u3} E (PseudoMetricSpace.toUniformSpace.{u3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))) _inst_8] {f : α -> 𝕜} {μ : MeasureTheory.Measure.{u1} α _inst_1} {c : E}, (Ne.{succ u3} E c (OfNat.ofNat.{u3} E 0 (OfNat.mk.{u3} E 0 (Zero.zero.{u3} E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (SubNegMonoid.toAddMonoid.{u3} E (AddGroup.toSubNegMonoid.{u3} E (NormedAddGroup.toAddGroup.{u3} E (NormedAddCommGroup.toNormedAddGroup.{u3} E _inst_6)))))))))) -> (Iff (AEMeasurable.{u1, u3} α E _inst_8 _inst_1 (fun (x : α) => SMul.smul.{u2, u3} 𝕜 E (SMulZeroClass.toHasSmul.{u2, u3} 𝕜 E (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} 𝕜 E (MulZeroClass.toHasZero.{u2} 𝕜 (MulZeroOneClass.toMulZeroClass.{u2} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2)))))) (AddZeroClass.toHasZero.{u3} E (AddMonoid.toAddZeroClass.{u3} E (AddCommMonoid.toAddMonoid.{u3} E (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6)))))) (Module.toMulActionWithZero.{u2, u3} 𝕜 E (Ring.toSemiring.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2))))) (AddCommGroup.toAddCommMonoid.{u3} E (SeminormedAddCommGroup.toAddCommGroup.{u3} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6))) (NormedSpace.toModule.{u2, u3} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} E _inst_6) _inst_7))))) (f x) c) μ) (AEMeasurable.{u1, u2} α 𝕜 _inst_4 _inst_1 f μ)) +but is expected to have type + forall {α : Type.{u3}} [_inst_1 : MeasurableSpace.{u3} α] {𝕜 : Type.{u1}} [_inst_2 : NontriviallyNormedField.{u1} 𝕜] [_inst_3 : CompleteSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))] [_inst_4 : MeasurableSpace.{u1} 𝕜] [_inst_5 : BorelSpace.{u1} 𝕜 (UniformSpace.toTopologicalSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))))) _inst_4] {E : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} E] [_inst_7 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_6)] [_inst_8 : MeasurableSpace.{u2} E] [_inst_9 : BorelSpace.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_6)))) _inst_8] {f : α -> 𝕜} {μ : MeasureTheory.Measure.{u3} α _inst_1} {c : E}, (Ne.{succ u2} E c (OfNat.ofNat.{u2} E 0 (Zero.toOfNat0.{u2} E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_6))))))))) -> (Iff (AEMeasurable.{u3, u2} α E _inst_8 _inst_1 (fun (x : α) => HSMul.hSMul.{u1, u2, u2} 𝕜 E E (instHSMul.{u1, u2} 𝕜 E (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_6)))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2)))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_6)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2)))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_6)))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2))))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_6)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_2) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_6) _inst_7)))))) (f x) c) μ) (AEMeasurable.{u3, u1} α 𝕜 _inst_4 _inst_1 f μ)) +Case conversion may be inaccurate. Consider using '#align ae_measurable_smul_const aemeasurable_smul_constₓ'. -/ +theorem aemeasurable_smul_const {f : α → 𝕜} {μ : Measure α} {c : E} (hc : c ≠ 0) : AEMeasurable (fun x => f x • c) μ ↔ AEMeasurable f μ := (closedEmbedding_smul_left hc).MeasurableEmbedding.aemeasurable_comp_iff -#align ae_measurable_smul_const aEMeasurable_smul_const +#align ae_measurable_smul_const aemeasurable_smul_const end NormedSpace diff --git a/Mathbin/MeasureTheory/Constructions/Polish.lean b/Mathbin/MeasureTheory/Constructions/Polish.lean index 2879bd8442..b86bfd264b 100644 --- a/Mathbin/MeasureTheory/Constructions/Polish.lean +++ b/Mathbin/MeasureTheory/Constructions/Polish.lean @@ -70,6 +70,7 @@ namespace MeasureTheory /-! ### Analytic sets -/ +#print MeasureTheory.AnalyticSet /- /-- An analytic set is a set which is the continuous image of some Polish space. There are several equivalent characterizations of this definition. For the definition, we pick one that avoids universe issues: a set is analytic if and only if it is a continuous image of `ℕ → ℕ` (or if it @@ -82,13 +83,17 @@ context of complex analysis. -/ irreducible_def AnalyticSet (s : Set α) : Prop := s = ∅ ∨ ∃ f : (ℕ → ℕ) → α, Continuous f ∧ range f = s #align measure_theory.analytic_set MeasureTheory.AnalyticSet +-/ +#print MeasureTheory.analyticSet_empty /- theorem analyticSet_empty : AnalyticSet (∅ : Set α) := by rw [analytic_set] exact Or.inl rfl #align measure_theory.analytic_set_empty MeasureTheory.analyticSet_empty +-/ +#print MeasureTheory.analyticSet_range_of_polishSpace /- theorem analyticSet_range_of_polishSpace {β : Type _} [TopologicalSpace β] [PolishSpace β] {f : β → α} (f_cont : Continuous f) : AnalyticSet (range f) := by @@ -101,7 +106,9 @@ theorem analyticSet_range_of_polishSpace {β : Type _} [TopologicalSpace β] [Po refine' Or.inr ⟨f ∘ g, f_cont.comp g_cont, _⟩ rwa [hg.range_comp] #align measure_theory.analytic_set_range_of_polish_space MeasureTheory.analyticSet_range_of_polishSpace +-/ +#print IsOpen.analyticSet_image /- /-- The image of an open set under a continuous map is analytic. -/ theorem IsOpen.analyticSet_image {β : Type _} [TopologicalSpace β] [PolishSpace β] {s : Set β} (hs : IsOpen s) {f : β → α} (f_cont : Continuous f) : AnalyticSet (f '' s) := @@ -110,7 +117,9 @@ theorem IsOpen.analyticSet_image {β : Type _} [TopologicalSpace β] [PolishSpac haveI : PolishSpace s := hs.polish_space exact analytic_set_range_of_polish_space (f_cont.comp continuous_subtype_val) #align is_open.analytic_set_image IsOpen.analyticSet_image +-/ +#print MeasureTheory.analyticSet_iff_exists_polishSpace_range /- /-- A set is analytic if and only if it is the continuous image of some Polish space. -/ theorem analyticSet_iff_exists_polishSpace_range {s : Set α} : AnalyticSet s ↔ @@ -130,7 +139,9 @@ theorem analyticSet_iff_exists_polishSpace_range {s : Set α} : rw [← f_range] exact analytic_set_range_of_polish_space f_cont #align measure_theory.analytic_set_iff_exists_polish_space_range MeasureTheory.analyticSet_iff_exists_polishSpace_range +-/ +#print MeasureTheory.AnalyticSet.image_of_continuousOn /- /-- The continuous image of an analytic set is analytic -/ theorem AnalyticSet.image_of_continuousOn {β : Type _} [TopologicalSpace β] {s : Set α} (hs : AnalyticSet s) {f : α → β} (hf : ContinuousOn f s) : AnalyticSet (f '' s) := @@ -144,12 +155,16 @@ theorem AnalyticSet.image_of_continuousOn {β : Type _} [TopologicalSpace β] {s rw [← gs] exact mem_range_self _ #align measure_theory.analytic_set.image_of_continuous_on MeasureTheory.AnalyticSet.image_of_continuousOn +-/ +#print MeasureTheory.AnalyticSet.image_of_continuous /- theorem AnalyticSet.image_of_continuous {β : Type _} [TopologicalSpace β] {s : Set α} (hs : AnalyticSet s) {f : α → β} (hf : Continuous f) : AnalyticSet (f '' s) := hs.image_of_continuousOn hf.ContinuousOn #align measure_theory.analytic_set.image_of_continuous MeasureTheory.AnalyticSet.image_of_continuous +-/ +#print MeasureTheory.AnalyticSet.iInter /- /-- A countable intersection of analytic sets is analytic. -/ theorem AnalyticSet.iInter [hι : Nonempty ι] [Countable ι] [T2Space α] {s : ι → Set α} (hs : ∀ n, AnalyticSet (s n)) : AnalyticSet (⋂ n, s n) := @@ -194,7 +209,9 @@ theorem AnalyticSet.iInter [hι : Nonempty ι] [Countable ι] [T2Space α] {s : rw [← F_range] exact analytic_set_range_of_polish_space F_cont #align measure_theory.analytic_set.Inter MeasureTheory.AnalyticSet.iInter +-/ +#print MeasureTheory.AnalyticSet.iUnion /- /-- A countable union of analytic sets is analytic. -/ theorem AnalyticSet.iUnion [Countable ι] {s : ι → Set α} (hs : ∀ n, AnalyticSet (s n)) : AnalyticSet (⋃ n, s n) := @@ -219,14 +236,18 @@ theorem AnalyticSet.iUnion [Countable ι] {s : ι → Set α} (hs : ∀ n, Analy rw [← F_range] exact analytic_set_range_of_polish_space F_cont #align measure_theory.analytic_set.Union MeasureTheory.AnalyticSet.iUnion +-/ +#print IsClosed.analyticSet /- theorem IsClosed.analyticSet [PolishSpace α] {s : Set α} (hs : IsClosed s) : AnalyticSet s := by haveI : PolishSpace s := hs.polish_space rw [← @Subtype.range_val α s] exact analytic_set_range_of_polish_space continuous_subtype_val #align is_closed.analytic_set IsClosed.analyticSet +-/ +#print MeasurableSet.isClopenable /- /-- Given a Borel-measurable set in a Polish space, there exists a finer Polish topology making it clopen. This is in fact an equivalence, see `is_clopenable_iff_measurable_set`. -/ theorem MeasurableSet.isClopenable [PolishSpace α] [MeasurableSpace α] [BorelSpace α] {s : Set α} @@ -238,7 +259,9 @@ theorem MeasurableSet.isClopenable [PolishSpace α] [MeasurableSpace α] [BorelS · exact fun u hu h'u => h'u.compl · exact fun f f_disj f_meas hf => is_clopenable.Union hf #align measurable_set.is_clopenable MeasurableSet.isClopenable +-/ +#print MeasurableSet.analyticSet /- theorem MeasurableSet.analyticSet {α : Type _} [t : TopologicalSpace α] [PolishSpace α] [MeasurableSpace α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s) : AnalyticSet s := by @@ -252,7 +275,14 @@ theorem MeasurableSet.analyticSet {α : Type _} [t : TopologicalSpace α] [Polis convert@analytic_set.image_of_continuous α t' α t s A id (continuous_id_of_le t't) simp only [id.def, image_id'] #align measurable_set.analytic_set MeasurableSet.analyticSet +-/ +/- warning: measurable.exists_continuous -> Measurable.exists_continuous is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [t : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α t] [_inst_3 : MeasurableSpace.{u1} α] [_inst_4 : BorelSpace.{u1} α t _inst_3] [tβ : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} β tβ] [_inst_6 : MeasurableSpace.{u2} β] [_inst_7 : BorelSpace.{u2} β tβ _inst_6] {f : α -> β}, (Measurable.{u1, u2} α β _inst_3 _inst_6 f) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' t) (And (Continuous.{u1, u2} α β t' tβ f) (PolishSpace.{u1} α t')))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [t : TopologicalSpace.{u2} α] [_inst_2 : PolishSpace.{u2} α t] [_inst_3 : MeasurableSpace.{u2} α] [_inst_4 : BorelSpace.{u2} α t _inst_3] [tβ : TopologicalSpace.{u1} β] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} β tβ] [_inst_6 : MeasurableSpace.{u1} β] [_inst_7 : BorelSpace.{u1} β tβ _inst_6] {f : α -> β}, (Measurable.{u2, u1} α β _inst_3 _inst_6 f) -> (Exists.{succ u2} (TopologicalSpace.{u2} α) (fun (t' : TopologicalSpace.{u2} α) => And (LE.le.{u2} (TopologicalSpace.{u2} α) (Preorder.toLE.{u2} (TopologicalSpace.{u2} α) (PartialOrder.toPreorder.{u2} (TopologicalSpace.{u2} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u2} α))) t' t) (And (Continuous.{u2, u1} α β t' tβ f) (PolishSpace.{u2} α t')))) +Case conversion may be inaccurate. Consider using '#align measurable.exists_continuous Measurable.exists_continuousₓ'. -/ /-- Given a Borel-measurable function from a Polish space to a second-countable space, there exists a finer Polish topology on the source space for which the function is continuous. -/ theorem Measurable.exists_continuous {α β : Type _} [t : TopologicalSpace α] [PolishSpace α] @@ -279,13 +309,21 @@ theorem Measurable.exists_continuous {α β : Type _} [t : TopologicalSpace α] /-! ### Separating sets with measurable sets -/ +#print MeasureTheory.MeasurablySeparable /- /-- Two sets `u` and `v` in a measurable space are measurably separable if there exists a measurable set containing `u` and disjoint from `v`. This is mostly interesting for Borel-separable sets. -/ def MeasurablySeparable {α : Type _} [MeasurableSpace α] (s t : Set α) : Prop := ∃ u, s ⊆ u ∧ Disjoint t u ∧ MeasurableSet u #align measure_theory.measurably_separable MeasureTheory.MeasurablySeparable +-/ +/- warning: measure_theory.measurably_separable.Union -> MeasureTheory.MeasurablySeparable.iUnion is a dubious translation: +lean 3 declaration is + forall {ι : Type.{u1}} [_inst_2 : Countable.{succ u1} ι] {α : Type.{u2}} [_inst_3 : MeasurableSpace.{u2} α] {s : ι -> (Set.{u2} α)} {t : ι -> (Set.{u2} α)}, (forall (m : ι) (n : ι), MeasureTheory.MeasurablySeparable.{u2} α _inst_3 (s m) (t n)) -> (MeasureTheory.MeasurablySeparable.{u2} α _inst_3 (Set.iUnion.{u2, succ u1} α ι (fun (n : ι) => s n)) (Set.iUnion.{u2, succ u1} α ι (fun (m : ι) => t m))) +but is expected to have type + forall {ι : Type.{u2}} [_inst_2 : Countable.{succ u2} ι] {α : Type.{u1}} [_inst_3 : MeasurableSpace.{u1} α] {s : ι -> (Set.{u1} α)} {t : ι -> (Set.{u1} α)}, (forall (m : ι) (n : ι), MeasureTheory.MeasurablySeparable.{u1} α _inst_3 (s m) (t n)) -> (MeasureTheory.MeasurablySeparable.{u1} α _inst_3 (Set.iUnion.{u1, succ u2} α ι (fun (n : ι) => s n)) (Set.iUnion.{u1, succ u2} α ι (fun (m : ι) => t m))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measurably_separable.Union MeasureTheory.MeasurablySeparable.iUnionₓ'. -/ theorem MeasurablySeparable.iUnion [Countable ι] {α : Type _} [MeasurableSpace α] {s t : ι → Set α} (h : ∀ m n, MeasurablySeparable (s m) (t n)) : MeasurablySeparable (⋃ n, s n) (⋃ m, t m) := by @@ -301,6 +339,12 @@ theorem MeasurablySeparable.iUnion [Countable ι] {α : Type _} [MeasurableSpace exact MeasurableSet.iInter fun n => hu m n #align measure_theory.measurably_separable.Union MeasureTheory.MeasurablySeparable.iUnion +/- warning: measure_theory.measurably_separable_range_of_disjoint -> MeasureTheory.measurablySeparable_range_of_disjoint is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : MeasurableSpace.{u1} α] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_3] {f : (Nat -> Nat) -> α} {g : (Nat -> Nat) -> α}, (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => Nat.topologicalSpace)) _inst_1 f) -> (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => Nat.topologicalSpace)) _inst_1 g) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Set.range.{u1, 1} α (Nat -> Nat) f) (Set.range.{u1, 1} α (Nat -> Nat) g)) -> (MeasureTheory.MeasurablySeparable.{u1} α _inst_3 (Set.range.{u1, 1} α (Nat -> Nat) f) (Set.range.{u1, 1} α (Nat -> Nat) g)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : MeasurableSpace.{u1} α] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_3] {f : (Nat -> Nat) -> α} {g : (Nat -> Nat) -> α}, (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => instTopologicalSpaceNat)) _inst_1 f) -> (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => instTopologicalSpaceNat)) _inst_1 g) -> (Disjoint.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Set.range.{u1, 1} α (Nat -> Nat) f) (Set.range.{u1, 1} α (Nat -> Nat) g)) -> (MeasureTheory.MeasurablySeparable.{u1} α _inst_3 (Set.range.{u1, 1} α (Nat -> Nat) f) (Set.range.{u1, 1} α (Nat -> Nat) g)) +Case conversion may be inaccurate. Consider using '#align measure_theory.measurably_separable_range_of_disjoint MeasureTheory.measurablySeparable_range_of_disjointₓ'. -/ /-- The hard part of the Lusin separation theorem saying that two disjoint analytic sets are contained in disjoint Borel sets (see the full statement in `analytic_set.measurably_separable`). Here, we prove this when our analytic sets are the ranges of functions from `ℕ → ℕ`. @@ -435,6 +479,12 @@ theorem measurablySeparable_range_of_disjoint [T2Space α] [MeasurableSpace α] exact M n B #align measure_theory.measurably_separable_range_of_disjoint MeasureTheory.measurablySeparable_range_of_disjoint +/- warning: measure_theory.analytic_set.measurably_separable -> MeasureTheory.AnalyticSet.measurablySeparable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : MeasurableSpace.{u1} α] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_3] {s : Set.{u1} α} {t : Set.{u1} α}, (MeasureTheory.AnalyticSet.{u1} α _inst_1 s) -> (MeasureTheory.AnalyticSet.{u1} α _inst_1 t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) s t) -> (MeasureTheory.MeasurablySeparable.{u1} α _inst_3 s t) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : MeasurableSpace.{u1} α] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_3] {s : Set.{u1} α} {t : Set.{u1} α}, (MeasureTheory.AnalyticSet.{u1} α _inst_1 s) -> (MeasureTheory.AnalyticSet.{u1} α _inst_1 t) -> (Disjoint.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) s t) -> (MeasureTheory.MeasurablySeparable.{u1} α _inst_3 s t) +Case conversion may be inaccurate. Consider using '#align measure_theory.analytic_set.measurably_separable MeasureTheory.AnalyticSet.measurablySeparableₓ'. -/ /-- The Lusin separation theorem: if two analytic sets are disjoint, then they are contained in disjoint Borel sets. -/ theorem AnalyticSet.measurablySeparable [T2Space α] [MeasurableSpace α] [BorelSpace α] {s t : Set α} @@ -455,6 +505,7 @@ variable {γ : Type _} [tγ : TopologicalSpace γ] [PolishSpace γ] include tγ +#print MeasureTheory.measurableSet_range_of_continuous_injective /- /-- The Lusin-Souslin theorem: the range of a continuous injective function defined on a Polish space is Borel-measurable. -/ theorem measurableSet_range_of_continuous_injective {β : Type _} [TopologicalSpace β] [T2Space β] @@ -618,7 +669,9 @@ theorem measurableSet_range_of_continuous_injective {β : Type _} [TopologicalSp -- the closure of `v`. exact disjoint_left.1 (hvw.closure_left w_open) this xw #align measure_theory.measurable_set_range_of_continuous_injective MeasureTheory.measurableSet_range_of_continuous_injective +-/ +#print IsClosed.measurableSet_image_of_continuousOn_injOn /- theorem IsClosed.measurableSet_image_of_continuousOn_injOn {β : Type _} [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [BorelSpace β] {s : Set γ} (hs : IsClosed s) {f : γ → β} (f_cont : ContinuousOn f s) (f_inj : InjOn f s) : MeasurableSet (f '' s) := @@ -629,12 +682,19 @@ theorem IsClosed.measurableSet_image_of_continuousOn_injOn {β : Type _} [Topolo · rwa [continuousOn_iff_continuous_restrict] at f_cont · rwa [inj_on_iff_injective] at f_inj #align is_closed.measurable_set_image_of_continuous_on_inj_on IsClosed.measurableSet_image_of_continuousOn_injOn +-/ variable [MeasurableSpace γ] [hγb : BorelSpace γ] {β : Type _} [tβ : TopologicalSpace β] [T2Space β] [MeasurableSpace β] [BorelSpace β] {s : Set γ} {f : γ → β} include tβ hγb +/- warning: measurable_set.image_of_continuous_on_inj_on -> MeasurableSet.image_of_continuousOn_injOn is a dubious translation: +lean 3 declaration is + forall {γ : Type.{u1}} [tγ : TopologicalSpace.{u1} γ] [_inst_2 : PolishSpace.{u1} γ tγ] [_inst_3 : MeasurableSpace.{u1} γ] [hγb : BorelSpace.{u1} γ tγ _inst_3] {β : Type.{u2}} [tβ : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β tβ] [_inst_5 : MeasurableSpace.{u2} β] [_inst_6 : BorelSpace.{u2} β tβ _inst_5] {s : Set.{u1} γ} {f : γ -> β}, (MeasurableSet.{u1} γ _inst_3 s) -> (ContinuousOn.{u1, u2} γ β tγ tβ f s) -> (Set.InjOn.{u1, u2} γ β f s) -> (MeasurableSet.{u2} β _inst_5 (Set.image.{u1, u2} γ β f s)) +but is expected to have type + forall {γ : Type.{u2}} [tγ : TopologicalSpace.{u2} γ] [_inst_2 : PolishSpace.{u2} γ tγ] [_inst_3 : MeasurableSpace.{u2} γ] [hγb : BorelSpace.{u2} γ tγ _inst_3] {β : Type.{u1}} [tβ : TopologicalSpace.{u1} β] [_inst_4 : T2Space.{u1} β tβ] [_inst_5 : MeasurableSpace.{u1} β] [_inst_6 : BorelSpace.{u1} β tβ _inst_5] {s : Set.{u2} γ} {f : γ -> β}, (MeasurableSet.{u2} γ _inst_3 s) -> (ContinuousOn.{u2, u1} γ β tγ tβ f s) -> (Set.InjOn.{u2, u1} γ β f s) -> (MeasurableSet.{u1} β _inst_5 (Set.image.{u2, u1} γ β f s)) +Case conversion may be inaccurate. Consider using '#align measurable_set.image_of_continuous_on_inj_on MeasurableSet.image_of_continuousOn_injOnₓ'. -/ /-- The Lusin-Souslin theorem: if `s` is Borel-measurable in a Polish space, then its image under a continuous injective map is also Borel-measurable. -/ theorem MeasurableSet.image_of_continuousOn_injOn (hs : MeasurableSet s) (f_cont : ContinuousOn f s) @@ -648,6 +708,7 @@ theorem MeasurableSet.image_of_continuousOn_injOn (hs : MeasurableSet s) (f_cont (f_cont.mono_dom t't) f_inj #align measurable_set.image_of_continuous_on_inj_on MeasurableSet.image_of_continuousOn_injOn +#print MeasurableSet.image_of_measurable_injOn /- /-- The Lusin-Souslin theorem: if `s` is Borel-measurable in a Polish space, then its image under a measurable injective map taking values in a second-countable topological space is also Borel-measurable. -/ @@ -673,7 +734,14 @@ theorem MeasurableSet.image_of_measurable_injOn [SecondCountableTopology β] (hs rfl) β _ _ _ _ s f M (@Continuous.continuousOn γ β t' tβ f s f_cont) f_inj #align measurable_set.image_of_measurable_inj_on MeasurableSet.image_of_measurable_injOn +-/ +/- warning: continuous.measurable_embedding -> Continuous.measurableEmbedding is a dubious translation: +lean 3 declaration is + forall {γ : Type.{u1}} [tγ : TopologicalSpace.{u1} γ] [_inst_2 : PolishSpace.{u1} γ tγ] [_inst_3 : MeasurableSpace.{u1} γ] [hγb : BorelSpace.{u1} γ tγ _inst_3] {β : Type.{u2}} [tβ : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β tβ] [_inst_5 : MeasurableSpace.{u2} β] [_inst_6 : BorelSpace.{u2} β tβ _inst_5] {f : γ -> β}, (Continuous.{u1, u2} γ β tγ tβ f) -> (Function.Injective.{succ u1, succ u2} γ β f) -> (MeasurableEmbedding.{u1, u2} γ β _inst_3 _inst_5 f) +but is expected to have type + forall {γ : Type.{u2}} [tγ : TopologicalSpace.{u2} γ] [_inst_2 : PolishSpace.{u2} γ tγ] [_inst_3 : MeasurableSpace.{u2} γ] [hγb : BorelSpace.{u2} γ tγ _inst_3] {β : Type.{u1}} [tβ : TopologicalSpace.{u1} β] [_inst_4 : T2Space.{u1} β tβ] [_inst_5 : MeasurableSpace.{u1} β] [_inst_6 : BorelSpace.{u1} β tβ _inst_5] {f : γ -> β}, (Continuous.{u2, u1} γ β tγ tβ f) -> (Function.Injective.{succ u2, succ u1} γ β f) -> (MeasurableEmbedding.{u2, u1} γ β _inst_3 _inst_5 f) +Case conversion may be inaccurate. Consider using '#align continuous.measurable_embedding Continuous.measurableEmbeddingₓ'. -/ /-- An injective continuous function on a Polish space is a measurable embedding. -/ theorem Continuous.measurableEmbedding (f_cont : Continuous f) (f_inj : Injective f) : MeasurableEmbedding f := @@ -683,6 +751,12 @@ theorem Continuous.measurableEmbedding (f_cont : Continuous f) (f_inj : Injectiv hu.image_of_continuousOn_injOn f_cont.ContinuousOn (f_inj.InjOn _) } #align continuous.measurable_embedding Continuous.measurableEmbedding +/- warning: continuous_on.measurable_embedding -> ContinuousOn.measurableEmbedding is a dubious translation: +lean 3 declaration is + forall {γ : Type.{u1}} [tγ : TopologicalSpace.{u1} γ] [_inst_2 : PolishSpace.{u1} γ tγ] [_inst_3 : MeasurableSpace.{u1} γ] [hγb : BorelSpace.{u1} γ tγ _inst_3] {β : Type.{u2}} [tβ : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β tβ] [_inst_5 : MeasurableSpace.{u2} β] [_inst_6 : BorelSpace.{u2} β tβ _inst_5] {s : Set.{u1} γ} {f : γ -> β}, (MeasurableSet.{u1} γ _inst_3 s) -> (ContinuousOn.{u1, u2} γ β tγ tβ f s) -> (Set.InjOn.{u1, u2} γ β f s) -> (MeasurableEmbedding.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} γ) Type.{u1} (Set.hasCoeToSort.{u1} γ) s) β (Subtype.instMeasurableSpace.{u1} γ (fun (x : γ) => Membership.Mem.{u1, u1} γ (Set.{u1} γ) (Set.hasMem.{u1} γ) x s) _inst_3) _inst_5 (Set.restrict.{u1, u2} γ (fun (ᾰ : γ) => β) s f)) +but is expected to have type + forall {γ : Type.{u2}} [tγ : TopologicalSpace.{u2} γ] [_inst_2 : PolishSpace.{u2} γ tγ] [_inst_3 : MeasurableSpace.{u2} γ] [hγb : BorelSpace.{u2} γ tγ _inst_3] {β : Type.{u1}} [tβ : TopologicalSpace.{u1} β] [_inst_4 : T2Space.{u1} β tβ] [_inst_5 : MeasurableSpace.{u1} β] [_inst_6 : BorelSpace.{u1} β tβ _inst_5] {s : Set.{u2} γ} {f : γ -> β}, (MeasurableSet.{u2} γ _inst_3 s) -> (ContinuousOn.{u2, u1} γ β tγ tβ f s) -> (Set.InjOn.{u2, u1} γ β f s) -> (MeasurableEmbedding.{u2, u1} (Set.Elem.{u2} γ s) β (Subtype.instMeasurableSpace.{u2} γ (fun (x : γ) => Membership.mem.{u2, u2} γ (Set.{u2} γ) (Set.instMembershipSet.{u2} γ) x s) _inst_3) _inst_5 (Set.restrict.{u2, u1} γ (fun (ᾰ : γ) => β) s f)) +Case conversion may be inaccurate. Consider using '#align continuous_on.measurable_embedding ContinuousOn.measurableEmbeddingₓ'. -/ /-- If `s` is Borel-measurable in a Polish space and `f` is continuous injective on `s`, then the restriction of `f` to `s` is a measurable embedding. -/ theorem ContinuousOn.measurableEmbedding (hs : MeasurableSet s) (f_cont : ContinuousOn f s) @@ -699,6 +773,7 @@ theorem ContinuousOn.measurableEmbedding (hs : MeasurableSet s) (f_cont : Contin rwa [← image_comp] at B } #align continuous_on.measurable_embedding ContinuousOn.measurableEmbedding +#print Measurable.measurableEmbedding /- /-- An injective measurable function from a Polish space to a second-countable topological space is a measurable embedding. -/ theorem Measurable.measurableEmbedding [SecondCountableTopology β] (f_meas : Measurable f) @@ -707,9 +782,11 @@ theorem Measurable.measurableEmbedding [SecondCountableTopology β] (f_meas : Me Measurable := f_meas measurableSet_image' := fun u hu => hu.image_of_measurable_injOn f_meas (f_inj.InjOn _) } #align measurable.measurable_embedding Measurable.measurableEmbedding +-/ omit tβ +#print MeasureTheory.isClopenable_iff_measurableSet /- /-- In a Polish space, a set is clopenable if and only if it is Borel-measurable. -/ theorem isClopenable_iff_measurableSet : IsClopenable s ↔ MeasurableSet s := by @@ -739,9 +816,16 @@ theorem isClopenable_iff_measurableSet : IsClopenable s ↔ MeasurableSet s := convert E.measurable_set_image.2 M simp only [id.def, image_id'] #align measure_theory.is_clopenable_iff_measurable_set MeasureTheory.isClopenable_iff_measurableSet +-/ omit hγb +/- warning: measure_theory.measurable_set_exists_tendsto -> MeasureTheory.measurableSet_exists_tendsto is a dubious translation: +lean 3 declaration is + forall {ι : Type.{u1}} {γ : Type.{u2}} [tγ : TopologicalSpace.{u2} γ] [_inst_2 : PolishSpace.{u2} γ tγ] [_inst_3 : MeasurableSpace.{u2} γ] {β : Type.{u3}} [_inst_5 : MeasurableSpace.{u3} β] [hγ : OpensMeasurableSpace.{u2} γ tγ _inst_3] [_inst_7 : Countable.{succ u1} ι] {l : Filter.{u1} ι} [_inst_8 : Filter.IsCountablyGenerated.{u1} ι l] {f : ι -> β -> γ}, (forall (i : ι), Measurable.{u3, u2} β γ _inst_5 _inst_3 (f i)) -> (MeasurableSet.{u3} β _inst_5 (setOf.{u3} β (fun (x : β) => Exists.{succ u2} γ (fun (c : γ) => Filter.Tendsto.{u1, u2} ι γ (fun (n : ι) => f n x) l (nhds.{u2} γ tγ c))))) +but is expected to have type + forall {ι : Type.{u2}} {γ : Type.{u3}} [tγ : TopologicalSpace.{u3} γ] [_inst_2 : PolishSpace.{u3} γ tγ] [_inst_3 : MeasurableSpace.{u3} γ] {β : Type.{u1}} [_inst_5 : MeasurableSpace.{u1} β] [hγ : OpensMeasurableSpace.{u3} γ tγ _inst_3] [_inst_7 : Countable.{succ u2} ι] {l : Filter.{u2} ι} [_inst_8 : Filter.IsCountablyGenerated.{u2} ι l] {f : ι -> β -> γ}, (forall (i : ι), Measurable.{u1, u3} β γ _inst_5 _inst_3 (f i)) -> (MeasurableSet.{u1} β _inst_5 (setOf.{u1} β (fun (x : β) => Exists.{succ u3} γ (fun (c : γ) => Filter.Tendsto.{u2, u3} ι γ (fun (n : ι) => f n x) l (nhds.{u3} γ tγ c))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measurable_set_exists_tendsto MeasureTheory.measurableSet_exists_tendstoₓ'. -/ /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/ /-- The set of points for which a measurable sequence of functions converges is measurable. -/ @[measurability] @@ -780,6 +864,7 @@ end MeasureTheory /-! ### The Borel Isomorphism Theorem -/ +#print polish_of_countable /- --Note: Move to topology/metric_space/polish when porting. instance (priority := 50) polish_of_countable [h : Countable α] [DiscreteTopology α] : PolishSpace α := by @@ -790,6 +875,7 @@ instance (priority := 50) polish_of_countable [h : Countable α] [DiscreteTopolo exact fun t _ => isClosed_discrete _ exact this.polish_space #align polish_of_countable polish_of_countable +-/ namespace PolishSpace @@ -808,13 +894,16 @@ variable {β : Type _} [TopologicalSpace β] [PolishSpace α] [PolishSpace β] variable [MeasurableSpace α] [MeasurableSpace β] [BorelSpace α] [BorelSpace β] +#print PolishSpace.borelSchroederBernstein /- /-- If two Polish spaces admit Borel measurable injections to one another, then they are Borel isomorphic.-/ noncomputable def borelSchroederBernstein {f : α → β} {g : β → α} (fmeas : Measurable f) (finj : Function.Injective f) (gmeas : Measurable g) (ginj : Function.Injective g) : α ≃ᵐ β := (fmeas.MeasurableEmbedding finj).schroeder_bernstein (gmeas.MeasurableEmbedding ginj) #align polish_space.borel_schroeder_bernstein PolishSpace.borelSchroederBernstein +-/ +#print PolishSpace.measurableEquivNatBoolOfNotCountable /- /-- Any uncountable Polish space is Borel isomorphic to the Cantor space `ℕ → bool`.-/ noncomputable def measurableEquivNatBoolOfNotCountable (h : ¬Countable α) : α ≃ᵐ (ℕ → Bool) := by @@ -825,12 +914,16 @@ noncomputable def measurableEquivNatBoolOfNotCountable (h : ¬Countable α) : α obtain ⟨g, gmeas, ginj⟩ := MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated α exact ⟨borel_schroeder_bernstein gmeas ginj fcts.measurable finj⟩ #align polish_space.measurable_equiv_nat_bool_of_not_countable PolishSpace.measurableEquivNatBoolOfNotCountable +-/ +#print PolishSpace.measurableEquivOfNotCountable /- /-- The **Borel Isomorphism Theorem**: Any two uncountable Polish spaces are Borel isomorphic.-/ noncomputable def measurableEquivOfNotCountable (hα : ¬Countable α) (hβ : ¬Countable β) : α ≃ᵐ β := (measurableEquivNatBoolOfNotCountable hα).trans (measurableEquivNatBoolOfNotCountable hβ).symm #align polish_space.measurable_equiv_of_not_countable PolishSpace.measurableEquivOfNotCountable +-/ +#print PolishSpace.Equiv.measurableEquiv /- /-- The **Borel Isomorphism Theorem**: If two Polish spaces have the same cardinality, they are Borel isomorphic.-/ noncomputable def Equiv.measurableEquiv (e : α ≃ β) : α ≃ᵐ β := @@ -842,6 +935,7 @@ noncomputable def Equiv.measurableEquiv (e : α ≃ β) : α ≃ᵐ β := refine' measurable_equiv_of_not_countable h _ rwa [e.countable_iff] at h #align polish_space.equiv.measurable_equiv PolishSpace.Equiv.measurableEquiv +-/ end PolishSpace @@ -853,6 +947,12 @@ instance [PolishSpace α] : PolishSpace (univ : Set α) := variable (α) [MeasurableSpace α] [PolishSpace α] [BorelSpace α] +/- warning: measure_theory.exists_nat_measurable_equiv_range_coe_fin_of_finite -> MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite is a dubious translation: +lean 3 declaration is + forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : PolishSpace.{u1} α _inst_1] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_2] [_inst_5 : Finite.{succ u1} α], Exists.{1} Nat (fun (n : Nat) => Nonempty.{succ u1} (MeasurableEquiv.{u1, 0} α (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.range.{0, 1} Real (Fin n) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Real (HasLiftT.mk.{1, 1} (Fin n) Real (CoeTCₓ.coe.{1, 1} (Fin n) Real (coeTrans.{1, 1, 1} (Fin n) Nat Real (Nat.castCoe.{0} Real Real.hasNatCast) (Fin.coeToNat n))))))) _inst_2 (Subtype.instMeasurableSpace.{0} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.range.{0, 1} Real (Fin n) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Fin n) Real (HasLiftT.mk.{1, 1} (Fin n) Real (CoeTCₓ.coe.{1, 1} (Fin n) Real (coeTrans.{1, 1, 1} (Fin n) Nat Real (Nat.castCoe.{0} Real Real.hasNatCast) (Fin.coeToNat n))))))) Real.measurableSpace))) +but is expected to have type + forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : PolishSpace.{u1} α _inst_1] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_2] [_inst_5 : Finite.{succ u1} α], Exists.{1} Nat (fun (n : Nat) => Nonempty.{succ u1} (MeasurableEquiv.{u1, 0} α (Set.Elem.{0} Real (Set.range.{0, 1} Real (Fin n) (fun (x : Fin n) => Nat.cast.{0} Real Real.natCast (Fin.val n x)))) _inst_2 (Subtype.instMeasurableSpace.{0} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.range.{0, 1} Real (Fin n) (fun (x : Fin n) => Nat.cast.{0} Real Real.natCast (Fin.val n x)))) Real.measurableSpace))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_nat_measurable_equiv_range_coe_fin_of_finite MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finiteₓ'. -/ theorem exists_nat_measurableEquiv_range_coe_fin_of_finite [Finite α] : ∃ n : ℕ, Nonempty (α ≃ᵐ range (coe : Fin n → ℝ)) := by @@ -861,6 +961,12 @@ theorem exists_nat_measurableEquiv_range_coe_fin_of_finite [Finite α] : exact Equiv.ofInjective _ (nat.cast_injective.comp Fin.val_injective) #align measure_theory.exists_nat_measurable_equiv_range_coe_fin_of_finite MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite +/- warning: measure_theory.measurable_equiv_range_coe_nat_of_infinite_of_countable -> MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable is a dubious translation: +lean 3 declaration is + forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : PolishSpace.{u1} α _inst_1] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_2] [_inst_5 : Infinite.{succ u1} α] [_inst_6 : Countable.{succ u1} α], Nonempty.{succ u1} (MeasurableEquiv.{u1, 0} α (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.range.{0, 1} Real Nat ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast)))))) _inst_2 (Subtype.instMeasurableSpace.{0} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x (Set.range.{0, 1} Real Nat ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Real (HasLiftT.mk.{1, 1} Nat Real (CoeTCₓ.coe.{1, 1} Nat Real (Nat.castCoe.{0} Real Real.hasNatCast)))))) Real.measurableSpace)) +but is expected to have type + forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : MeasurableSpace.{u1} α] [_inst_3 : PolishSpace.{u1} α _inst_1] [_inst_4 : BorelSpace.{u1} α _inst_1 _inst_2] [_inst_5 : Infinite.{succ u1} α] [_inst_6 : Countable.{succ u1} α], Nonempty.{succ u1} (MeasurableEquiv.{u1, 0} α (Set.Elem.{0} Real (Set.range.{0, 1} Real Nat (Nat.cast.{0} Real Real.natCast))) _inst_2 (Subtype.instMeasurableSpace.{0} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x (Set.range.{0, 1} Real Nat (Nat.cast.{0} Real Real.natCast))) Real.measurableSpace)) +Case conversion may be inaccurate. Consider using '#align measure_theory.measurable_equiv_range_coe_nat_of_infinite_of_countable MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countableₓ'. -/ theorem measurableEquiv_range_coe_nat_of_infinite_of_countable [Infinite α] [Countable α] : Nonempty (α ≃ᵐ range (coe : ℕ → ℝ)) := by @@ -871,6 +977,7 @@ theorem measurableEquiv_range_coe_nat_of_infinite_of_countable [Infinite α] [Co exact Equiv.ofInjective coe Nat.cast_injective #align measure_theory.measurable_equiv_range_coe_nat_of_infinite_of_countable MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable +#print MeasureTheory.exists_subset_real_measurableEquiv /- /-- Any Polish Borel space is measurably equivalent to a subset of the reals. -/ theorem exists_subset_real_measurableEquiv : ∃ s : Set ℝ, MeasurableSet s ∧ Nonempty (α ≃ᵐ s) := by @@ -896,13 +1003,16 @@ theorem exists_subset_real_measurableEquiv : ∃ s : Set ℝ, MeasurableSet s rw [countable_coe_iff] exact Cardinal.not_countable_real #align measure_theory.exists_subset_real_measurable_equiv MeasureTheory.exists_subset_real_measurableEquiv +-/ +#print MeasureTheory.exists_measurableEmbedding_real /- /-- Any Polish Borel space embeds measurably into the reals. -/ theorem exists_measurableEmbedding_real : ∃ f : α → ℝ, MeasurableEmbedding f := by obtain ⟨s, hs, ⟨e⟩⟩ := exists_subset_real_measurable_equiv α exact ⟨coe ∘ e, (MeasurableEmbedding.subtype_coe hs).comp e.measurable_embedding⟩ #align measure_theory.exists_measurable_embedding_real MeasureTheory.exists_measurableEmbedding_real +-/ end MeasureTheory diff --git a/Mathbin/MeasureTheory/Constructions/Prod/Basic.lean b/Mathbin/MeasureTheory/Constructions/Prod/Basic.lean index 851f8bc6b5..832f3eafd8 100644 --- a/Mathbin/MeasureTheory/Constructions/Prod/Basic.lean +++ b/Mathbin/MeasureTheory/Constructions/Prod/Basic.lean @@ -227,7 +227,7 @@ theorem Measurable.lintegral_prod_right' [SigmaFinite ν] : ∀ {f : α × β → ℝ≥0∞} (hf : Measurable f), Measurable fun x => ∫⁻ y, f (x, y) ∂ν := by have m := @measurable_prod_mk_left - refine' Measurable.eNNReal_induction _ _ _ + refine' Measurable.ennreal_induction _ _ _ · intro c s hs simp only [← indicator_comp_right] suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator _ (m hs)] @@ -723,7 +723,7 @@ theorem lintegral_prod_of_measurable : ∀ (f : α × β → ℝ≥0∞) (hf : Measurable f), (∫⁻ z, f z ∂μ.Prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ := by have m := @measurable_prod_mk_left - refine' Measurable.eNNReal_induction _ _ _ + refine' Measurable.ennreal_induction _ _ _ · intro c s hs simp only [← indicator_comp_right] simp [lintegral_indicator, m hs, hs, lintegral_const_mul, measurable_measure_prod_mk_left hs, diff --git a/Mathbin/MeasureTheory/Covering/Differentiation.lean b/Mathbin/MeasureTheory/Covering/Differentiation.lean index 42a4144f13..f4c70e58d7 100644 --- a/Mathbin/MeasureTheory/Covering/Differentiation.lean +++ b/Mathbin/MeasureTheory/Covering/Differentiation.lean @@ -478,7 +478,7 @@ theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) : theorem aEMeasurable_limRatio : AEMeasurable (v.limRatio ρ) μ := by - apply ENNReal.aEMeasurable_of_exist_almost_disjoint_supersets _ _ fun p q hpq => _ + apply ENNReal.aemeasurable_of_exist_almost_disjoint_supersets _ _ fun p q hpq => _ exact v.exists_measurable_supersets_lim_ratio hρ hpq #align vitali_family.ae_measurable_lim_ratio VitaliFamily.aEMeasurable_limRatio diff --git a/Mathbin/MeasureTheory/Function/AeMeasurableOrder.lean b/Mathbin/MeasureTheory/Function/AeMeasurableOrder.lean index becd0e25b1..11056de548 100644 --- a/Mathbin/MeasureTheory/Function/AeMeasurableOrder.lean +++ b/Mathbin/MeasureTheory/Function/AeMeasurableOrder.lean @@ -30,10 +30,16 @@ open MeasureTheory Set TopologicalSpace open Classical ENNReal NNReal +/- warning: measure_theory.ae_measurable_of_exist_almost_disjoint_supersets -> MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {β : Type.{u2}} [_inst_1 : CompleteLinearOrder.{u2} β] [_inst_2 : DenselyOrdered.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β (CompleteLinearOrder.toCompleteLattice.{u2} β _inst_1)))))] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : OrderTopology.{u2} β _inst_3 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β (CompleteLinearOrder.toCompleteLattice.{u2} β _inst_1))))] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u2} β _inst_3] [_inst_6 : MeasurableSpace.{u2} β] [_inst_7 : BorelSpace.{u2} β _inst_3 _inst_6] (s : Set.{u2} β), (Set.Countable.{u2} β s) -> (Dense.{u2} β _inst_3 s) -> (forall (f : α -> β), (forall (p : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) p s) -> (forall (q : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) q s) -> (LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β (CompleteLinearOrder.toCompleteLattice.{u2} β _inst_1))))) p q) -> (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (MeasurableSet.{u1} α m u) (And (MeasurableSet.{u1} α m v) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β (CompleteLinearOrder.toCompleteLattice.{u2} β _inst_1))))) (f x) p)) u) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{u2} β (Preorder.toHasLt.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β (CompleteLinearOrder.toCompleteLattice.{u2} β _inst_1))))) q (f x))) v) (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u v)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))))))))))) -> (AEMeasurable.{u1, u2} α β _inst_6 m f μ)) +but is expected to have type + forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} (μ : MeasureTheory.Measure.{u2} α m) {β : Type.{u1}} [_inst_1 : CompleteLinearOrder.{u1} β] [_inst_2 : DenselyOrdered.{u1} β (Preorder.toLT.{u1} β (PartialOrder.toPreorder.{u1} β (OmegaCompletePartialOrder.toPartialOrder.{u1} β (CompleteLattice.instOmegaCompletePartialOrder.{u1} β (CompleteLinearOrder.toCompleteLattice.{u1} β _inst_1)))))] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : OrderTopology.{u1} β _inst_3 (PartialOrder.toPreorder.{u1} β (OmegaCompletePartialOrder.toPartialOrder.{u1} β (CompleteLattice.instOmegaCompletePartialOrder.{u1} β (CompleteLinearOrder.toCompleteLattice.{u1} β _inst_1))))] [_inst_5 : TopologicalSpace.SecondCountableTopology.{u1} β _inst_3] [_inst_6 : MeasurableSpace.{u1} β] [_inst_7 : BorelSpace.{u1} β _inst_3 _inst_6] (s : Set.{u1} β), (Set.Countable.{u1} β s) -> (Dense.{u1} β _inst_3 s) -> (forall (f : α -> β), (forall (p : β), (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) p s) -> (forall (q : β), (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) q s) -> (LT.lt.{u1} β (Preorder.toLT.{u1} β (PartialOrder.toPreorder.{u1} β (OmegaCompletePartialOrder.toPartialOrder.{u1} β (CompleteLattice.instOmegaCompletePartialOrder.{u1} β (CompleteLinearOrder.toCompleteLattice.{u1} β _inst_1))))) p q) -> (Exists.{succ u2} (Set.{u2} α) (fun (u : Set.{u2} α) => Exists.{succ u2} (Set.{u2} α) (fun (v : Set.{u2} α) => And (MeasurableSet.{u2} α m u) (And (MeasurableSet.{u2} α m v) (And (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (setOf.{u2} α (fun (x : α) => LT.lt.{u1} β (Preorder.toLT.{u1} β (PartialOrder.toPreorder.{u1} β (OmegaCompletePartialOrder.toPartialOrder.{u1} β (CompleteLattice.instOmegaCompletePartialOrder.{u1} β (CompleteLinearOrder.toCompleteLattice.{u1} β _inst_1))))) (f x) p)) u) (And (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (setOf.{u2} α (fun (x : α) => LT.lt.{u1} β (Preorder.toLT.{u1} β (PartialOrder.toPreorder.{u1} β (OmegaCompletePartialOrder.toPartialOrder.{u1} β (CompleteLattice.instOmegaCompletePartialOrder.{u1} β (CompleteLinearOrder.toCompleteLattice.{u1} β _inst_1))))) q (f x))) v) (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m μ) (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) u v)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))))))))))) -> (AEMeasurable.{u2, u1} α β _inst_6 m f μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_measurable_of_exist_almost_disjoint_supersets MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersetsₓ'. -/ /-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable supersets which are disjoint up to measure zero when `p < q`, then `f` is almost-everywhere measurable. It is even enough to have this for `p` and `q` in a countable dense set. -/ -theorem MeasureTheory.aEMeasurable_of_exist_almost_disjoint_supersets {α : Type _} +theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type _} {m : MeasurableSpace α} (μ : Measure α) {β : Type _} [CompleteLinearOrder β] [DenselyOrdered β] [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β] [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β) @@ -123,12 +129,18 @@ theorem MeasureTheory.aEMeasurable_of_exist_almost_disjoint_supersets {α : Type have A : x ∈ u' r := mem_bInter fun i hi => (huv r i).2.2.1 xr simp only [A, rq, piecewise_eq_of_mem, Subtype.coe_mk] exact ⟨f', f'_meas, ff'⟩ -#align measure_theory.ae_measurable_of_exist_almost_disjoint_supersets MeasureTheory.aEMeasurable_of_exist_almost_disjoint_supersets +#align measure_theory.ae_measurable_of_exist_almost_disjoint_supersets MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets +/- warning: ennreal.ae_measurable_of_exist_almost_disjoint_supersets -> ENNReal.aemeasurable_of_exist_almost_disjoint_supersets is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) (f : α -> ENNReal), (forall (p : NNReal) (q : NNReal), (LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) p q) -> (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (MeasurableSet.{u1} α m u) (And (MeasurableSet.{u1} α m v) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) p))) u) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) q) (f x))) v) (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) u v)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))))))))))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) (f : α -> ENNReal), (forall (p : NNReal) (q : NNReal), (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) p q) -> (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (MeasurableSet.{u1} α m u) (And (MeasurableSet.{u1} α m v) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (ENNReal.some p))) u) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.some q) (f x))) v) (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) u v)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)))))))))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) +Case conversion may be inaccurate. Consider using '#align ennreal.ae_measurable_of_exist_almost_disjoint_supersets ENNReal.aemeasurable_of_exist_almost_disjoint_supersetsₓ'. -/ /-- If a function `f : α → ℝ≥0∞` is such that the level sets `{f < p}` and `{q < f}` have measurable supersets which are disjoint up to measure zero when `p` and `q` are finite numbers satisfying `p < q`, then `f` is almost-everywhere measurable. -/ -theorem ENNReal.aEMeasurable_of_exist_almost_disjoint_supersets {α : Type _} {m : MeasurableSpace α} +theorem ENNReal.aemeasurable_of_exist_almost_disjoint_supersets {α : Type _} {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) (h : ∀ (p : ℝ≥0) (q : ℝ≥0), @@ -143,10 +155,10 @@ theorem ENNReal.aEMeasurable_of_exist_almost_disjoint_supersets {α : Type _} {m ∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ∞ ∉ s := ENNReal.exists_countable_dense_no_zero_top have I : ∀ x ∈ s, x ≠ ∞ := fun x xs hx => s_top (hx ▸ xs) - apply MeasureTheory.aEMeasurable_of_exist_almost_disjoint_supersets μ s s_count s_dense _ + apply MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets μ s s_count s_dense _ rintro p hp q hq hpq lift p to ℝ≥0 using I p hp lift q to ℝ≥0 using I q hq exact h p q (ENNReal.coe_lt_coe.1 hpq) -#align ennreal.ae_measurable_of_exist_almost_disjoint_supersets ENNReal.aEMeasurable_of_exist_almost_disjoint_supersets +#align ennreal.ae_measurable_of_exist_almost_disjoint_supersets ENNReal.aemeasurable_of_exist_almost_disjoint_supersets diff --git a/Mathbin/MeasureTheory/Function/Floor.lean b/Mathbin/MeasureTheory/Function/Floor.lean index 8d5a366ab4..049d68a3af 100644 --- a/Mathbin/MeasureTheory/Function/Floor.lean +++ b/Mathbin/MeasureTheory/Function/Floor.lean @@ -25,47 +25,61 @@ section FloorRing variable {α R : Type _} [MeasurableSpace α] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] +#print Int.measurable_floor /- theorem Int.measurable_floor [OpensMeasurableSpace R] : Measurable (Int.floor : R → ℤ) := measurable_to_countable fun x => by simpa only [Int.preimage_floor_singleton] using measurableSet_Ico #align int.measurable_floor Int.measurable_floor +-/ +#print Measurable.floor /- @[measurability] theorem Measurable.floor [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) : Measurable fun x => ⌊f x⌋ := Int.measurable_floor.comp hf #align measurable.floor Measurable.floor +-/ +#print Int.measurable_ceil /- theorem Int.measurable_ceil [OpensMeasurableSpace R] : Measurable (Int.ceil : R → ℤ) := measurable_to_countable fun x => by simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc #align int.measurable_ceil Int.measurable_ceil +-/ +#print Measurable.ceil /- @[measurability] theorem Measurable.ceil [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) : Measurable fun x => ⌈f x⌉ := Int.measurable_ceil.comp hf #align measurable.ceil Measurable.ceil +-/ +#print measurable_fract /- theorem measurable_fract [BorelSpace R] : Measurable (Int.fract : R → R) := by intro s hs rw [Int.preimage_fract] exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico) #align measurable_fract measurable_fract +-/ +#print Measurable.fract /- @[measurability] theorem Measurable.fract [BorelSpace R] {f : α → R} (hf : Measurable f) : Measurable fun x => Int.fract (f x) := measurable_fract.comp hf #align measurable.fract Measurable.fract +-/ +#print MeasurableSet.image_fract /- theorem MeasurableSet.image_fract [BorelSpace R] {s : Set R} (hs : MeasurableSet s) : MeasurableSet (Int.fract '' s) := by simp only [Int.image_fract, sub_eq_add_neg, image_add_right'] exact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico #align measurable_set.image_fract MeasurableSet.image_fract +-/ end FloorRing @@ -74,21 +88,37 @@ section FloorSemiring variable {α R : Type _} [MeasurableSpace α] [LinearOrderedSemiring R] [FloorSemiring R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R] {f : α → R} +#print Nat.measurable_floor /- theorem Nat.measurable_floor : Measurable (Nat.floor : R → ℕ) := measurable_to_countable fun n => by cases eq_or_ne ⌊n⌋₊ 0 <;> simp [*, Nat.preimage_floor_of_ne_zero] #align nat.measurable_floor Nat.measurable_floor +-/ +/- warning: measurable.nat_floor -> Measurable.nat_floor is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {R : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : LinearOrderedSemiring.{u2} R] [_inst_3 : FloorSemiring.{u2} R (StrictOrderedSemiring.toOrderedSemiring.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2))] [_inst_4 : TopologicalSpace.{u2} R] [_inst_5 : OrderTopology.{u2} R _inst_4 (PartialOrder.toPreorder.{u2} R (OrderedCancelAddCommMonoid.toPartialOrder.{u2} R (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2))))] [_inst_6 : MeasurableSpace.{u2} R] [_inst_7 : OpensMeasurableSpace.{u2} R _inst_4 _inst_6] {f : α -> R}, (Measurable.{u1, u2} α R _inst_1 _inst_6 f) -> (Measurable.{u1, 0} α Nat _inst_1 Nat.instMeasurableSpace (fun (x : α) => Nat.floor.{u2} R (StrictOrderedSemiring.toOrderedSemiring.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2)) _inst_3 (f x))) +but is expected to have type + forall {α : Type.{u2}} {R : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : LinearOrderedSemiring.{u1} R] [_inst_3 : FloorSemiring.{u1} R (StrictOrderedSemiring.toOrderedSemiring.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2))] [_inst_4 : TopologicalSpace.{u1} R] [_inst_5 : OrderTopology.{u1} R _inst_4 (PartialOrder.toPreorder.{u1} R (StrictOrderedSemiring.toPartialOrder.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2)))] [_inst_6 : MeasurableSpace.{u1} R] [_inst_7 : OpensMeasurableSpace.{u1} R _inst_4 _inst_6] {f : α -> R}, (Measurable.{u2, u1} α R _inst_1 _inst_6 f) -> (Measurable.{u2, 0} α Nat _inst_1 Nat.instMeasurableSpace (fun (x : α) => Nat.floor.{u1} R (StrictOrderedSemiring.toOrderedSemiring.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2)) _inst_3 (f x))) +Case conversion may be inaccurate. Consider using '#align measurable.nat_floor Measurable.nat_floorₓ'. -/ @[measurability] theorem Measurable.nat_floor (hf : Measurable f) : Measurable fun x => ⌊f x⌋₊ := Nat.measurable_floor.comp hf #align measurable.nat_floor Measurable.nat_floor +#print Nat.measurable_ceil /- theorem Nat.measurable_ceil : Measurable (Nat.ceil : R → ℕ) := measurable_to_countable fun n => by cases eq_or_ne ⌈n⌉₊ 0 <;> simp [*, Nat.preimage_ceil_of_ne_zero] #align nat.measurable_ceil Nat.measurable_ceil +-/ +/- warning: measurable.nat_ceil -> Measurable.nat_ceil is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {R : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : LinearOrderedSemiring.{u2} R] [_inst_3 : FloorSemiring.{u2} R (StrictOrderedSemiring.toOrderedSemiring.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2))] [_inst_4 : TopologicalSpace.{u2} R] [_inst_5 : OrderTopology.{u2} R _inst_4 (PartialOrder.toPreorder.{u2} R (OrderedCancelAddCommMonoid.toPartialOrder.{u2} R (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2))))] [_inst_6 : MeasurableSpace.{u2} R] [_inst_7 : OpensMeasurableSpace.{u2} R _inst_4 _inst_6] {f : α -> R}, (Measurable.{u1, u2} α R _inst_1 _inst_6 f) -> (Measurable.{u1, 0} α Nat _inst_1 Nat.instMeasurableSpace (fun (x : α) => Nat.ceil.{u2} R (StrictOrderedSemiring.toOrderedSemiring.{u2} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} R _inst_2)) _inst_3 (f x))) +but is expected to have type + forall {α : Type.{u2}} {R : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : LinearOrderedSemiring.{u1} R] [_inst_3 : FloorSemiring.{u1} R (StrictOrderedSemiring.toOrderedSemiring.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2))] [_inst_4 : TopologicalSpace.{u1} R] [_inst_5 : OrderTopology.{u1} R _inst_4 (PartialOrder.toPreorder.{u1} R (StrictOrderedSemiring.toPartialOrder.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2)))] [_inst_6 : MeasurableSpace.{u1} R] [_inst_7 : OpensMeasurableSpace.{u1} R _inst_4 _inst_6] {f : α -> R}, (Measurable.{u2, u1} α R _inst_1 _inst_6 f) -> (Measurable.{u2, 0} α Nat _inst_1 Nat.instMeasurableSpace (fun (x : α) => Nat.ceil.{u1} R (StrictOrderedSemiring.toOrderedSemiring.{u1} R (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} R _inst_2)) _inst_3 (f x))) +Case conversion may be inaccurate. Consider using '#align measurable.nat_ceil Measurable.nat_ceilₓ'. -/ @[measurability] theorem Measurable.nat_ceil (hf : Measurable f) : Measurable fun x => ⌈f x⌉₊ := Nat.measurable_ceil.comp hf diff --git a/Mathbin/MeasureTheory/Function/SimpleFunc.lean b/Mathbin/MeasureTheory/Function/SimpleFunc.lean index 00c32431f0..e89c412af8 100644 --- a/Mathbin/MeasureTheory/Function/SimpleFunc.lean +++ b/Mathbin/MeasureTheory/Function/SimpleFunc.lean @@ -41,6 +41,7 @@ namespace MeasureTheory variable {α β γ δ : Type _} +#print MeasureTheory.SimpleFunc /- /-- A function `f` from a measurable space to any type is called *simple*, if every preimage `f ⁻¹' {x}` is measurable, and the range is finite. This structure bundles a function with these properties. -/ @@ -49,6 +50,7 @@ structure SimpleFunc.{u, v} (α : Type u) [MeasurableSpace α] (β : Type v) whe measurableSet_fiber' : ∀ x, MeasurableSet (to_fun ⁻¹' {x}) finite_range' : (Set.range to_fun).Finite #align measure_theory.simple_func MeasureTheory.SimpleFunc +-/ -- mathport name: «expr →ₛ » local infixr:25 " →ₛ " => SimpleFunc @@ -59,107 +61,211 @@ section Measurable variable [MeasurableSpace α] -instance hasCoeToFun : CoeFun (α →ₛ β) fun _ => α → β := +#print MeasureTheory.SimpleFunc.instCoeFun /- +instance instCoeFun : CoeFun (α →ₛ β) fun _ => α → β := ⟨toFun⟩ -#align measure_theory.simple_func.has_coe_to_fun MeasureTheory.SimpleFunc.hasCoeToFun +#align measure_theory.simple_func.has_coe_to_fun MeasureTheory.SimpleFunc.instCoeFun +-/ +/- warning: measure_theory.simple_func.coe_injective -> MeasureTheory.SimpleFunc.coe_injective is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {{f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}} {{g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}}, (Eq.{max (succ u1) (succ u2)} ((fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) f) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g)) -> (Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) f g) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {{f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β}} {{g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β}}, (Eq.{max (succ u2) (succ u1)} (α -> β) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β g)) -> (Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) f g) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_injective MeasureTheory.SimpleFunc.coe_injectiveₓ'. -/ theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g := by cases f <;> cases g <;> congr <;> exact H #align measure_theory.simple_func.coe_injective MeasureTheory.SimpleFunc.coe_injective +/- warning: measure_theory.simple_func.ext -> MeasureTheory.SimpleFunc.ext is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, (forall (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g a)) -> (Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) f g) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β}, (forall (a : α), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f a) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β g a)) -> (Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) f g) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.ext MeasureTheory.SimpleFunc.extₓ'. -/ @[ext] theorem ext {f g : α →ₛ β} (H : ∀ a, f a = g a) : f = g := coe_injective <| funext H #align measure_theory.simple_func.ext MeasureTheory.SimpleFunc.ext +/- warning: measure_theory.simple_func.finite_range -> MeasureTheory.SimpleFunc.finite_range is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Set.Finite.{u2} β (Set.range.{u2, succ u1} β α (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Set.Finite.{u1} β (Set.range.{u1, succ u2} β α (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.finite_range MeasureTheory.SimpleFunc.finite_rangeₓ'. -/ theorem finite_range (f : α →ₛ β) : (Set.range f).Finite := f.finite_range' #align measure_theory.simple_func.finite_range MeasureTheory.SimpleFunc.finite_range +/- warning: measure_theory.simple_func.measurable_set_fiber -> MeasureTheory.SimpleFunc.measurableSet_fiber is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (x : β), MeasurableSet.{u1} α _inst_1 (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) x)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (x : β), MeasurableSet.{u2} α _inst_1 (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) x)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.measurable_set_fiber MeasureTheory.SimpleFunc.measurableSet_fiberₓ'. -/ theorem measurableSet_fiber (f : α →ₛ β) (x : β) : MeasurableSet (f ⁻¹' {x}) := f.measurableSet_fiber' x #align measure_theory.simple_func.measurable_set_fiber MeasureTheory.SimpleFunc.measurableSet_fiber +/- warning: measure_theory.simple_func.apply_mk -> MeasureTheory.SimpleFunc.apply_mk is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> β) (h : forall (x : β), MeasurableSet.{u1} α _inst_1 (Set.preimage.{u1, u2} α β f (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) x))) (h' : Set.Finite.{u2} β (Set.range.{u2, succ u1} β α f)) (x : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.mk.{u1, u2} α _inst_1 β f h h') x) (f x) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : α -> β) (h : forall (x : β), MeasurableSet.{u2} α _inst_1 (Set.preimage.{u2, u1} α β f (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) x))) (h' : Set.Finite.{u1} β (Set.range.{u1, succ u2} β α f)) (x : α), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.mk.{u2, u1} α _inst_1 β f h h') x) (f x) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.apply_mk MeasureTheory.SimpleFunc.apply_mkₓ'. -/ @[simp] theorem apply_mk (f : α → β) (h h') (x : α) : SimpleFunc.mk f h h' x = f x := rfl #align measure_theory.simple_func.apply_mk MeasureTheory.SimpleFunc.apply_mk +#print MeasureTheory.SimpleFunc.ofIsEmpty /- /-- Simple function defined on the empty type. -/ def ofIsEmpty [IsEmpty α] : α →ₛ β where toFun := isEmptyElim measurableSet_fiber' x := Subsingleton.measurableSet finite_range' := by simp [range_eq_empty] #align measure_theory.simple_func.of_is_empty MeasureTheory.SimpleFunc.ofIsEmpty +-/ +#print MeasureTheory.SimpleFunc.range /- /-- Range of a simple function `α →ₛ β` as a `finset β`. -/ protected def range (f : α →ₛ β) : Finset β := f.finite_range.toFinset #align measure_theory.simple_func.range MeasureTheory.SimpleFunc.range +-/ +/- warning: measure_theory.simple_func.mem_range -> MeasureTheory.SimpleFunc.mem_range is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {b : β}, Iff (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) b (Set.range.{u2, succ u1} β α (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {b : β}, Iff (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) b (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) b (Set.range.{u1, succ u2} β α (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.mem_range MeasureTheory.SimpleFunc.mem_rangeₓ'. -/ @[simp] theorem mem_range {f : α →ₛ β} {b} : b ∈ f.range ↔ b ∈ range f := Finite.mem_toFinset _ #align measure_theory.simple_func.mem_range MeasureTheory.SimpleFunc.mem_range +/- warning: measure_theory.simple_func.mem_range_self -> MeasureTheory.SimpleFunc.mem_range_self is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (x : α), Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f x) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (x : α), Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f x) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.mem_range_self MeasureTheory.SimpleFunc.mem_range_selfₓ'. -/ theorem mem_range_self (f : α →ₛ β) (x : α) : f x ∈ f.range := mem_range.2 ⟨x, rfl⟩ #align measure_theory.simple_func.mem_range_self MeasureTheory.SimpleFunc.mem_range_self +/- warning: measure_theory.simple_func.coe_range -> MeasureTheory.SimpleFunc.coe_range is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{succ u2} (Set.{u2} β) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) (Set.range.{u2, succ u1} β α (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{succ u1} (Set.{u1} β) (Finset.toSet.{u1} β (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) (Set.range.{u1, succ u2} β α (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_range MeasureTheory.SimpleFunc.coe_rangeₓ'. -/ @[simp] theorem coe_range (f : α →ₛ β) : (↑f.range : Set β) = Set.range f := f.finite_range.coe_toFinset #align measure_theory.simple_func.coe_range MeasureTheory.SimpleFunc.coe_range +/- warning: measure_theory.simple_func.mem_range_of_measure_ne_zero -> MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {x : β} {μ : MeasureTheory.Measure.{u1} α _inst_1}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) x))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) x (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {x : β} {μ : MeasureTheory.Measure.{u2} α _inst_1}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α _inst_1 μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) x))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) x (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.mem_range_of_measure_ne_zero MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zeroₓ'. -/ theorem mem_range_of_measure_ne_zero {f : α →ₛ β} {x : β} {μ : Measure α} (H : μ (f ⁻¹' {x}) ≠ 0) : x ∈ f.range := let ⟨a, ha⟩ := nonempty_of_measure_ne_zero H mem_range.2 ⟨a, ha⟩ #align measure_theory.simple_func.mem_range_of_measure_ne_zero MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero +/- warning: measure_theory.simple_func.forall_range_iff -> MeasureTheory.SimpleFunc.forall_range_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {p : β -> Prop}, Iff (forall (y : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) y (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) -> (p y)) (forall (x : α), p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f x)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {p : β -> Prop}, Iff (forall (y : β), (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) y (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) -> (p y)) (forall (x : α), p (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f x)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.forall_range_iff MeasureTheory.SimpleFunc.forall_range_iffₓ'. -/ theorem forall_range_iff {f : α →ₛ β} {p : β → Prop} : (∀ y ∈ f.range, p y) ↔ ∀ x, p (f x) := by simp only [mem_range, Set.forall_range_iff] #align measure_theory.simple_func.forall_range_iff MeasureTheory.SimpleFunc.forall_range_iff +/- warning: measure_theory.simple_func.exists_range_iff -> MeasureTheory.SimpleFunc.exists_range_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {p : β -> Prop}, Iff (Exists.{succ u2} β (fun (y : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) y (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) (fun (H : Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) y (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) => p y))) (Exists.{succ u1} α (fun (x : α) => p (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f x))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {p : β -> Prop}, Iff (Exists.{succ u1} β (fun (y : β) => And (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) y (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) (p y))) (Exists.{succ u2} α (fun (x : α) => p (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.exists_range_iff MeasureTheory.SimpleFunc.exists_range_iffₓ'. -/ theorem exists_range_iff {f : α →ₛ β} {p : β → Prop} : (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x) := by simpa only [mem_range, exists_prop] using Set.exists_range_iff #align measure_theory.simple_func.exists_range_iff MeasureTheory.SimpleFunc.exists_range_iff +/- warning: measure_theory.simple_func.preimage_eq_empty_iff -> MeasureTheory.SimpleFunc.preimage_eq_empty_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (b : β), Iff (Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (Not (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (b : β), Iff (Eq.{succ u2} (Set.{u2} α) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) b)) (EmptyCollection.emptyCollection.{u2} (Set.{u2} α) (Set.instEmptyCollectionSet.{u2} α))) (Not (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) b (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.preimage_eq_empty_iff MeasureTheory.SimpleFunc.preimage_eq_empty_iffₓ'. -/ theorem preimage_eq_empty_iff (f : α →ₛ β) (b : β) : f ⁻¹' {b} = ∅ ↔ b ∉ f.range := preimage_singleton_eq_empty.trans <| not_congr mem_range.symm #align measure_theory.simple_func.preimage_eq_empty_iff MeasureTheory.SimpleFunc.preimage_eq_empty_iff +/- warning: measure_theory.simple_func.exists_forall_le -> MeasureTheory.SimpleFunc.exists_forall_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : Preorder.{u2} β] [_inst_4 : IsDirected.{u2} β (LE.le.{u2} β (Preorder.toHasLe.{u2} β _inst_3))] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Exists.{succ u2} β (fun (C : β) => forall (x : α), LE.le.{u2} β (Preorder.toHasLe.{u2} β _inst_3) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f x) C) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : Preorder.{u2} β] [_inst_4 : IsDirected.{u2} β (fun (x._@.Mathlib.MeasureTheory.Function.SimpleFunc._hyg.1377 : β) (x._@.Mathlib.MeasureTheory.Function.SimpleFunc._hyg.1379 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_3) x._@.Mathlib.MeasureTheory.Function.SimpleFunc._hyg.1377 x._@.Mathlib.MeasureTheory.Function.SimpleFunc._hyg.1379)] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Exists.{succ u2} β (fun (C : β) => forall (x : α), LE.le.{u2} β (Preorder.toLE.{u2} β _inst_3) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β f x) C) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.exists_forall_le MeasureTheory.SimpleFunc.exists_forall_leₓ'. -/ theorem exists_forall_le [Nonempty β] [Preorder β] [IsDirected β (· ≤ ·)] (f : α →ₛ β) : ∃ C, ∀ x, f x ≤ C := f.range.exists_le.imp fun C => forall_range_iff.1 #align measure_theory.simple_func.exists_forall_le MeasureTheory.SimpleFunc.exists_forall_le +#print MeasureTheory.SimpleFunc.const /- /-- Constant function as a `simple_func`. -/ def const (α) {β} [MeasurableSpace α] (b : β) : α →ₛ β := ⟨fun a => b, fun x => MeasurableSet.const _, finite_range_const⟩ #align measure_theory.simple_func.const MeasureTheory.SimpleFunc.const +-/ instance [Inhabited β] : Inhabited (α →ₛ β) := ⟨const _ default⟩ +#print MeasureTheory.SimpleFunc.const_apply /- theorem const_apply (a : α) (b : β) : (const α b) a = b := rfl #align measure_theory.simple_func.const_apply MeasureTheory.SimpleFunc.const_apply +-/ +/- warning: measure_theory.simple_func.coe_const -> MeasureTheory.SimpleFunc.coe_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (b : β), Eq.{max (succ u1) (succ u2)} (α -> β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 b)) (Function.const.{succ u2, succ u1} β α b) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (b : β), Eq.{max (succ u2) (succ u1)} (α -> β) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.const.{u2, u1} α β _inst_1 b)) (Function.const.{succ u1, succ u2} β α b) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_const MeasureTheory.SimpleFunc.coe_constₓ'. -/ @[simp] theorem coe_const (b : β) : ⇑(const α b) = Function.const α b := rfl #align measure_theory.simple_func.coe_const MeasureTheory.SimpleFunc.coe_const +#print MeasureTheory.SimpleFunc.range_const /- @[simp] theorem range_const (α) [MeasurableSpace α] [Nonempty α] (b : β) : (const α b).range = {b} := Finset.coe_injective <| by simp #align measure_theory.simple_func.range_const MeasureTheory.SimpleFunc.range_const +-/ +#print MeasureTheory.SimpleFunc.range_const_subset /- theorem range_const_subset (α) [MeasurableSpace α] (b : β) : (const α b).range ⊆ {b} := Finset.coe_subset.1 <| by simp #align measure_theory.simple_func.range_const_subset MeasureTheory.SimpleFunc.range_const_subset +-/ +/- warning: measure_theory.simple_func.simple_func_bot -> MeasureTheory.SimpleFunc.simpleFunc_bot is a dubious translation: +lean 3 declaration is + forall {β : Type.{u1}} {α : Type.{u2}} (f : MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) β) [_inst_2 : Nonempty.{succ u1} β], Exists.{succ u1} β (fun (c : β) => forall (x : α), Eq.{succ u1} β (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) β) (fun (_x : MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u1} α β (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α)))) f x) c) +but is expected to have type + forall {β : Type.{u1}} {α : Type.{u2}} (f : MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u2} α))) β) [_inst_2 : Nonempty.{succ u1} β], Exists.{succ u1} β (fun (c : β) => forall (x : α), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u2} α))) β f x) c) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.simple_func_bot MeasureTheory.SimpleFunc.simpleFunc_botₓ'. -/ theorem simpleFunc_bot {α} (f : @SimpleFunc α ⊥ β) [Nonempty β] : ∃ c, ∀ x, f x = c := by have hf_meas := @simple_func.measurable_set_fiber α _ ⊥ f @@ -179,6 +285,12 @@ theorem simpleFunc_bot {α} (f : @SimpleFunc α ⊥ β) [Nonempty β] : ∃ c, rwa [Set.mem_preimage, Set.mem_singleton_iff] at this #align measure_theory.simple_func.simple_func_bot MeasureTheory.SimpleFunc.simpleFunc_bot +/- warning: measure_theory.simple_func.simple_func_bot' -> MeasureTheory.SimpleFunc.simpleFunc_bot' is a dubious translation: +lean 3 declaration is + forall {β : Type.{u1}} {α : Type.{u2}} [_inst_2 : Nonempty.{succ u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) β), Exists.{succ u1} β (fun (c : β) => Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) β) f (MeasureTheory.SimpleFunc.const.{u2, u1} α β (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toHasBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.completeLattice.{u2} α))) c)) +but is expected to have type + forall {β : Type.{u1}} {α : Type.{u2}} [_inst_2 : Nonempty.{succ u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u2} α))) β), Exists.{succ u1} β (fun (c : β) => Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u2, u1} α (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u2} α))) β) f (MeasureTheory.SimpleFunc.const.{u2, u1} α β (Bot.bot.{u2} (MeasurableSpace.{u2} α) (CompleteLattice.toBot.{u2} (MeasurableSpace.{u2} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u2} α))) c)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.simple_func_bot' MeasureTheory.SimpleFunc.simpleFunc_bot'ₓ'. -/ theorem simpleFunc_bot' {α} [Nonempty β] (f : @SimpleFunc α ⊥ β) : ∃ c, f = @SimpleFunc.const α _ ⊥ c := by @@ -188,6 +300,12 @@ theorem simpleFunc_bot' {α} [Nonempty β] (f : @SimpleFunc α ⊥ β) : rw [h_eq x, simple_func.coe_const] #align measure_theory.simple_func.simple_func_bot' MeasureTheory.SimpleFunc.simpleFunc_bot' +/- warning: measure_theory.simple_func.measurable_set_cut -> MeasureTheory.SimpleFunc.measurableSet_cut is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (r : α -> β -> Prop) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), (forall (b : β), MeasurableSet.{u1} α _inst_1 (setOf.{u1} α (fun (a : α) => r a b))) -> (MeasurableSet.{u1} α _inst_1 (setOf.{u1} α (fun (a : α) => r a (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (r : α -> β -> Prop) (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), (forall (b : β), MeasurableSet.{u2} α _inst_1 (setOf.{u2} α (fun (a : α) => r a b))) -> (MeasurableSet.{u2} α _inst_1 (setOf.{u2} α (fun (a : α) => r a (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.measurable_set_cut MeasureTheory.SimpleFunc.measurableSet_cutₓ'. -/ theorem measurableSet_cut (r : α → β → Prop) (f : α →ₛ β) (h : ∀ b, MeasurableSet { a | r a b }) : MeasurableSet { a | r a (f a) } := by @@ -202,33 +320,56 @@ theorem measurableSet_cut (r : α → β → Prop) (f : α →ₛ β) (h : ∀ b MeasurableSet.inter (h b) (f.measurable_set_fiber _) #align measure_theory.simple_func.measurable_set_cut MeasureTheory.SimpleFunc.measurableSet_cut +/- warning: measure_theory.simple_func.measurable_set_preimage -> MeasureTheory.SimpleFunc.measurableSet_preimage is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (s : Set.{u2} β), MeasurableSet.{u1} α _inst_1 (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) s) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (s : Set.{u1} β), MeasurableSet.{u2} α _inst_1 (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) s) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.measurable_set_preimage MeasureTheory.SimpleFunc.measurableSet_preimageₓ'. -/ @[measurability] theorem measurableSet_preimage (f : α →ₛ β) (s) : MeasurableSet (f ⁻¹' s) := measurableSet_cut (fun _ b => b ∈ s) f fun b => MeasurableSet.const (b ∈ s) #align measure_theory.simple_func.measurable_set_preimage MeasureTheory.SimpleFunc.measurableSet_preimage +#print MeasureTheory.SimpleFunc.measurable /- /-- A simple function is measurable -/ @[measurability] protected theorem measurable [MeasurableSpace β] (f : α →ₛ β) : Measurable f := fun s _ => measurableSet_preimage f s #align measure_theory.simple_func.measurable MeasureTheory.SimpleFunc.measurable +-/ +#print MeasureTheory.SimpleFunc.aemeasurable /- @[measurability] -protected theorem aEMeasurable [MeasurableSpace β] {μ : Measure α} (f : α →ₛ β) : +protected theorem aemeasurable [MeasurableSpace β] {μ : Measure α} (f : α →ₛ β) : AEMeasurable f μ := f.Measurable.AEMeasurable -#align measure_theory.simple_func.ae_measurable MeasureTheory.SimpleFunc.aEMeasurable +#align measure_theory.simple_func.ae_measurable MeasureTheory.SimpleFunc.aemeasurable +-/ +/- warning: measure_theory.simple_func.sum_measure_preimage_singleton -> MeasureTheory.SimpleFunc.sum_measure_preimage_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) {μ : MeasureTheory.Measure.{u1} α _inst_1} (s : Finset.{u2} β), Eq.{1} ENNReal (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (y : β) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) y)))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) s))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) {μ : MeasureTheory.Measure.{u2} α _inst_1} (s : Finset.{u1} β), Eq.{1} ENNReal (Finset.sum.{0, u1} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (y : β) => MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α _inst_1 μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) y)))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α _inst_1 μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Finset.toSet.{u1} β s))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.sum_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_measure_preimage_singletonₓ'. -/ protected theorem sum_measure_preimage_singleton (f : α →ₛ β) {μ : Measure α} (s : Finset β) : (∑ y in s, μ (f ⁻¹' {y})) = μ (f ⁻¹' ↑s) := sum_measure_preimage_singleton _ fun _ _ => f.measurableSet_fiber _ #align measure_theory.simple_func.sum_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_measure_preimage_singleton +/- warning: measure_theory.simple_func.sum_range_measure_preimage_singleton -> MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (μ : MeasureTheory.Measure.{u1} α _inst_1), Eq.{1} ENNReal (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f) (fun (y : β) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) y)))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (Set.univ.{u1} α)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (μ : MeasureTheory.Measure.{u2} α _inst_1), Eq.{1} ENNReal (Finset.sum.{0, u1} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f) (fun (y : β) => MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α _inst_1 μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) y)))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α _inst_1 μ) (Set.univ.{u2} α)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.sum_range_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_range_measure_preimage_singletonₓ'. -/ theorem sum_range_measure_preimage_singleton (f : α →ₛ β) (μ : Measure α) : (∑ y in f.range, μ (f ⁻¹' {y})) = μ univ := by rw [f.sum_measure_preimage_singleton, coe_range, preimage_range] #align measure_theory.simple_func.sum_range_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton +#print MeasureTheory.SimpleFunc.piecewise /- /-- If-then-else as a `simple_func`. -/ def piecewise (s : Set α) (hs : MeasurableSet s) (f g : α →ₛ β) : α →ₛ β := ⟨s.piecewise f g, fun x => @@ -236,39 +377,82 @@ def piecewise (s : Set α) (hs : MeasurableSet s) (f g : α →ₛ β) : α → f.measurable.piecewise hs g.measurable trivial, (f.finite_range.union g.finite_range).Subset range_ite_subset⟩ #align measure_theory.simple_func.piecewise MeasureTheory.SimpleFunc.piecewise +-/ +/- warning: measure_theory.simple_func.coe_piecewise -> MeasureTheory.SimpleFunc.coe_piecewise is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (α -> β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s hs f g)) (Set.piecewise.{u1, succ u2} α (fun (ᾰ : α) => β) s (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g) (fun (j : α) => Classical.propDecidable (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) j s))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {s : Set.{u2} α} (hs : MeasurableSet.{u2} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (α -> β) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 s hs f g)) (Set.piecewise.{u2, succ u1} α (fun (ᾰ : α) => β) s (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β g) (fun (j : α) => Classical.propDecidable (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) j s))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_piecewise MeasureTheory.SimpleFunc.coe_piecewiseₓ'. -/ @[simp] theorem coe_piecewise {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) : ⇑(piecewise s hs f g) = s.piecewise f g := rfl #align measure_theory.simple_func.coe_piecewise MeasureTheory.SimpleFunc.coe_piecewise +/- warning: measure_theory.simple_func.piecewise_apply -> MeasureTheory.SimpleFunc.piecewise_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s hs f g) a) (ite.{succ u2} β (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (Classical.propDecidable (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g a)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {s : Set.{u2} α} (hs : MeasurableSet.{u2} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (a : α), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 s hs f g) a) (ite.{succ u1} β (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a s) (Classical.propDecidable (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a s)) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f a) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β g a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.piecewise_apply MeasureTheory.SimpleFunc.piecewise_applyₓ'. -/ theorem piecewise_apply {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) (a) : piecewise s hs f g a = if a ∈ s then f a else g a := rfl #align measure_theory.simple_func.piecewise_apply MeasureTheory.SimpleFunc.piecewise_apply +/- warning: measure_theory.simple_func.piecewise_compl -> MeasureTheory.SimpleFunc.piecewise_compl is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s)) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s) hs f g) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s (MeasurableSet.of_compl.{u1} α s _inst_1 hs) g f) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {s : Set.{u2} α} (hs : MeasurableSet.{u2} α _inst_1 (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) s)) (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 (HasCompl.compl.{u2} (Set.{u2} α) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} α) (Set.instBooleanAlgebraSet.{u2} α)) s) hs f g) (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 s (MeasurableSet.of_compl.{u2} α s _inst_1 hs) g f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.piecewise_compl MeasureTheory.SimpleFunc.piecewise_complₓ'. -/ @[simp] theorem piecewise_compl {s : Set α} (hs : MeasurableSet (sᶜ)) (f g : α →ₛ β) : piecewise (sᶜ) hs f g = piecewise s hs.ofCompl g f := coe_injective <| by simp [hs] #align measure_theory.simple_func.piecewise_compl MeasureTheory.SimpleFunc.piecewise_compl +/- warning: measure_theory.simple_func.piecewise_univ -> MeasureTheory.SimpleFunc.piecewise_univ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 (Set.univ.{u1} α) (MeasurableSet.univ.{u1} α _inst_1) f g) f +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 (Set.univ.{u2} α) (MeasurableSet.univ.{u2} α _inst_1) f g) f +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.piecewise_univ MeasureTheory.SimpleFunc.piecewise_univₓ'. -/ @[simp] theorem piecewise_univ (f g : α →ₛ β) : piecewise univ MeasurableSet.univ f g = f := coe_injective <| by simp #align measure_theory.simple_func.piecewise_univ MeasureTheory.SimpleFunc.piecewise_univ +/- warning: measure_theory.simple_func.piecewise_empty -> MeasureTheory.SimpleFunc.piecewise_empty is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)) (MeasurableSet.empty.{u1} α _inst_1) f g) g +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 (EmptyCollection.emptyCollection.{u2} (Set.{u2} α) (Set.instEmptyCollectionSet.{u2} α)) (MeasurableSet.empty.{u2} α _inst_1) f g) g +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.piecewise_empty MeasureTheory.SimpleFunc.piecewise_emptyₓ'. -/ @[simp] theorem piecewise_empty (f g : α →ₛ β) : piecewise ∅ MeasurableSet.empty f g = g := coe_injective <| by simp #align measure_theory.simple_func.piecewise_empty MeasureTheory.SimpleFunc.piecewise_empty +/- warning: measure_theory.simple_func.support_indicator -> MeasureTheory.SimpleFunc.support_indicator is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{succ u1} (Set.{u1} α) (Function.support.{u1, u2} α β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s hs f (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2))))))) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Function.support.{u1, u2} α β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{succ u1} (Set.{u1} α) (Function.support.{u1, u2} α β _inst_2 (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s hs f (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_2)))))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (Function.support.{u1, u2} α β _inst_2 (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β f))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.support_indicator MeasureTheory.SimpleFunc.support_indicatorₓ'. -/ theorem support_indicator [Zero β] {s : Set α} (hs : MeasurableSet s) (f : α →ₛ β) : Function.support (f.piecewise s hs (SimpleFunc.const α 0)) = s ∩ Function.support f := Set.support_indicator #align measure_theory.simple_func.support_indicator MeasureTheory.SimpleFunc.support_indicator +/- warning: measure_theory.simple_func.range_indicator -> MeasureTheory.SimpleFunc.range_indicator is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s), (Set.Nonempty.{u1} α s) -> (Ne.{succ u1} (Set.{u1} α) s (Set.univ.{u1} α)) -> (forall (x : β) (y : β), Eq.{succ u2} (Finset.{u2} β) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α β _inst_1 s hs (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 x) (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 y))) (Insert.insert.{u2, u2} β (Finset.{u2} β) (Finset.hasInsert.{u2} β (fun (a : β) (b : β) => Classical.propDecidable (Eq.{succ u2} β a b))) x (Singleton.singleton.{u2, u2} β (Finset.{u2} β) (Finset.hasSingleton.{u2} β) y))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] {s : Set.{u2} α} (hs : MeasurableSet.{u2} α _inst_1 s), (Set.Nonempty.{u2} α s) -> (Ne.{succ u2} (Set.{u2} α) s (Set.univ.{u2} α)) -> (forall (x : β) (y : β), Eq.{succ u1} (Finset.{u1} β) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α β _inst_1 s hs (MeasureTheory.SimpleFunc.const.{u2, u1} α β _inst_1 x) (MeasureTheory.SimpleFunc.const.{u2, u1} α β _inst_1 y))) (Insert.insert.{u1, u1} β (Finset.{u1} β) (Finset.instInsertFinset.{u1} β (fun (a : β) (b : β) => Classical.propDecidable (Eq.{succ u1} β a b))) x (Singleton.singleton.{u1, u1} β (Finset.{u1} β) (Finset.instSingletonFinset.{u1} β) y))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.range_indicator MeasureTheory.SimpleFunc.range_indicatorₓ'. -/ theorem range_indicator {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Nonempty) (hs_ne_univ : s ≠ univ) (x y : β) : (piecewise s hs (const α x) (const α y)).range = {x, y} := by @@ -277,11 +461,18 @@ theorem range_indicator {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Non (nonempty_compl.2 hs_ne_univ).image_const, singleton_union] #align measure_theory.simple_func.range_indicator MeasureTheory.SimpleFunc.range_indicator +/- warning: measure_theory.simple_func.measurable_bind -> MeasureTheory.SimpleFunc.measurable_bind is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u3} γ] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : β -> α -> γ), (forall (b : β), Measurable.{u1, u3} α γ _inst_1 _inst_2 (g b)) -> (Measurable.{u1, u3} α γ _inst_1 _inst_2 (fun (a : α) => g (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a) a)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u3} γ] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (g : β -> α -> γ), (forall (b : β), Measurable.{u2, u3} α γ _inst_1 _inst_2 (g b)) -> (Measurable.{u2, u3} α γ _inst_1 _inst_2 (fun (a : α) => g (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f a) a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.measurable_bind MeasureTheory.SimpleFunc.measurable_bindₓ'. -/ theorem measurable_bind [MeasurableSpace γ] (f : α →ₛ β) (g : β → α → γ) (hg : ∀ b, Measurable (g b)) : Measurable fun a => g (f a) a := fun s hs => f.measurableSet_cut (fun a b => g b a ∈ s) fun b => hg b hs #align measure_theory.simple_func.measurable_bind MeasureTheory.SimpleFunc.measurable_bind +#print MeasureTheory.SimpleFunc.bind /- /-- If `f : α →ₛ β` is a simple function and `g : β → α →ₛ γ` is a family of simple functions, then `f.bind g` binds the first argument of `g` to `f`. In other words, `f.bind g a = g (f a) a`. -/ def bind (f : α →ₛ β) (g : β → α →ₛ γ) : α →ₛ γ := @@ -290,41 +481,86 @@ def bind (f : α →ₛ β) (g : β → α →ₛ γ) : α →ₛ γ := (f.finite_range.biUnion fun b _ => (g b).finite_range).Subset <| by rintro _ ⟨a, rfl⟩ <;> simp <;> exact ⟨a, a, rfl⟩⟩ #align measure_theory.simple_func.bind MeasureTheory.SimpleFunc.bind +-/ +/- warning: measure_theory.simple_func.bind_apply -> MeasureTheory.SimpleFunc.bind_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : β -> (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ)) (a : α), Eq.{succ u3} γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.bind.{u1, u2, u3} α β γ _inst_1 f g) a) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (g (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a)) a) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : β -> (MeasureTheory.SimpleFunc.{u3, u1} α _inst_1 γ)) (a : α), Eq.{succ u1} γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.bind.{u3, u2, u1} α β γ _inst_1 f g) a) (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (g (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f a)) a) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.bind_apply MeasureTheory.SimpleFunc.bind_applyₓ'. -/ @[simp] theorem bind_apply (f : α →ₛ β) (g : β → α →ₛ γ) (a) : f.bind g a = g (f a) a := rfl #align measure_theory.simple_func.bind_apply MeasureTheory.SimpleFunc.bind_apply +#print MeasureTheory.SimpleFunc.map /- /-- Given a function `g : β → γ` and a simple function `f : α →ₛ β`, `f.map g` return the simple function `g ∘ f : α →ₛ γ` -/ def map (g : β → γ) (f : α →ₛ β) : α →ₛ γ := bind f (const α ∘ g) #align measure_theory.simple_func.map MeasureTheory.SimpleFunc.map +-/ +/- warning: measure_theory.simple_func.map_apply -> MeasureTheory.SimpleFunc.map_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (a : α), Eq.{succ u3} γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f) a) (g (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (a : α), Eq.{succ u1} γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.map.{u3, u2, u1} α β γ _inst_1 g f) a) (g (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_apply MeasureTheory.SimpleFunc.map_applyₓ'. -/ theorem map_apply (g : β → γ) (f : α →ₛ β) (a) : f.map g a = g (f a) := rfl #align measure_theory.simple_func.map_apply MeasureTheory.SimpleFunc.map_apply +/- warning: measure_theory.simple_func.map_map -> MeasureTheory.SimpleFunc.map_map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {δ : Type.{u4}} [_inst_1 : MeasurableSpace.{u1} α] (g : β -> γ) (h : γ -> δ) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u4)} (MeasureTheory.SimpleFunc.{u1, u4} α _inst_1 δ) (MeasureTheory.SimpleFunc.map.{u1, u3, u4} α γ δ _inst_1 h (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) (MeasureTheory.SimpleFunc.map.{u1, u2, u4} α β δ _inst_1 (Function.comp.{succ u2, succ u3, succ u4} β γ δ h g) f) +but is expected to have type + forall {α : Type.{u4}} {β : Type.{u3}} {γ : Type.{u1}} {δ : Type.{u2}} [_inst_1 : MeasurableSpace.{u4} α] (g : β -> γ) (h : γ -> δ) (f : MeasureTheory.SimpleFunc.{u4, u3} α _inst_1 β), Eq.{max (succ u4) (succ u2)} (MeasureTheory.SimpleFunc.{u4, u2} α _inst_1 δ) (MeasureTheory.SimpleFunc.map.{u4, u1, u2} α γ δ _inst_1 h (MeasureTheory.SimpleFunc.map.{u4, u3, u1} α β γ _inst_1 g f)) (MeasureTheory.SimpleFunc.map.{u4, u3, u2} α β δ _inst_1 (Function.comp.{succ u3, succ u1, succ u2} β γ δ h g) f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_map MeasureTheory.SimpleFunc.map_mapₓ'. -/ theorem map_map (g : β → γ) (h : γ → δ) (f : α →ₛ β) : (f.map g).map h = f.map (h ∘ g) := rfl #align measure_theory.simple_func.map_map MeasureTheory.SimpleFunc.map_map +/- warning: measure_theory.simple_func.coe_map -> MeasureTheory.SimpleFunc.coe_map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u3)} ((fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) (Function.comp.{succ u1, succ u2, succ u3} α β γ g (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β), Eq.{max (succ u3) (succ u1)} (α -> γ) (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.map.{u3, u2, u1} α β γ _inst_1 g f)) (Function.comp.{succ u3, succ u2, succ u1} α β γ g (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_map MeasureTheory.SimpleFunc.coe_mapₓ'. -/ @[simp] theorem coe_map (g : β → γ) (f : α →ₛ β) : (f.map g : α → γ) = g ∘ f := rfl #align measure_theory.simple_func.coe_map MeasureTheory.SimpleFunc.coe_map +/- warning: measure_theory.simple_func.range_map -> MeasureTheory.SimpleFunc.range_map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : DecidableEq.{succ u3} γ] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{succ u3} (Finset.{u3} γ) (MeasureTheory.SimpleFunc.range.{u1, u3} α γ _inst_1 (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) (Finset.image.{u2, u3} β γ (fun (a : γ) (b : γ) => _inst_2 a b) g (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : DecidableEq.{succ u3} γ] (g : β -> γ) (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{succ u3} (Finset.{u3} γ) (MeasureTheory.SimpleFunc.range.{u2, u3} α γ _inst_1 (MeasureTheory.SimpleFunc.map.{u2, u1, u3} α β γ _inst_1 g f)) (Finset.image.{u1, u3} β γ (fun (a : γ) (b : γ) => _inst_2 a b) g (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.range_map MeasureTheory.SimpleFunc.range_mapₓ'. -/ @[simp] theorem range_map [DecidableEq γ] (g : β → γ) (f : α →ₛ β) : (f.map g).range = f.range.image g := Finset.coe_injective <| by simp only [coe_range, coe_map, Finset.coe_image, range_comp] #align measure_theory.simple_func.range_map MeasureTheory.SimpleFunc.range_map +/- warning: measure_theory.simple_func.map_const -> MeasureTheory.SimpleFunc.map_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (g : β -> γ) (b : β), Eq.{max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1 b)) (MeasureTheory.SimpleFunc.const.{u1, u3} α γ _inst_1 (g b)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u1}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u3} α] (g : β -> γ) (b : β), Eq.{max (succ u3) (succ u2)} (MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.map.{u3, u1, u2} α β γ _inst_1 g (MeasureTheory.SimpleFunc.const.{u3, u1} α β _inst_1 b)) (MeasureTheory.SimpleFunc.const.{u3, u2} α γ _inst_1 (g b)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_const MeasureTheory.SimpleFunc.map_constₓ'. -/ @[simp] theorem map_const (g : β → γ) (b : β) : (const α b).map g = const α (g b) := rfl #align measure_theory.simple_func.map_const MeasureTheory.SimpleFunc.map_const +/- warning: measure_theory.simple_func.map_preimage -> MeasureTheory.SimpleFunc.map_preimage is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : β -> γ) (s : Set.{u3} γ), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u3} α γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) s) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) (Finset.filter.{u2} β (fun (b : β) => Membership.Mem.{u3, u3} γ (Set.{u3} γ) (Set.hasMem.{u3} γ) (g b) s) (fun (a : β) => Classical.propDecidable ((fun (b : β) => Membership.Mem.{u3, u3} γ (Set.{u3} γ) (Set.hasMem.{u3} γ) (g b) s) a)) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)))) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : β -> γ) (s : Set.{u1} γ), Eq.{succ u3} (Set.{u3} α) (Set.preimage.{u3, u1} α γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.map.{u3, u2, u1} α β γ _inst_1 g f)) s) (Set.preimage.{u3, u2} α β (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f) (Finset.toSet.{u2} β (Finset.filter.{u2} β (fun (b : β) => Membership.mem.{u1, u1} γ (Set.{u1} γ) (Set.instMembershipSet.{u1} γ) (g b) s) (fun (a : β) => Classical.propDecidable ((fun (b : β) => Membership.mem.{u1, u1} γ (Set.{u1} γ) (Set.instMembershipSet.{u1} γ) (g b) s) a)) (MeasureTheory.SimpleFunc.range.{u3, u2} α β _inst_1 f)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_preimage MeasureTheory.SimpleFunc.map_preimageₓ'. -/ theorem map_preimage (f : α →ₛ β) (g : β → γ) (s : Set γ) : f.map g ⁻¹' s = f ⁻¹' ↑(f.range.filterₓ fun b => g b ∈ s) := by @@ -333,11 +569,18 @@ theorem map_preimage (f : α →ₛ β) (g : β → γ) (s : Set γ) : apply preimage_comp #align measure_theory.simple_func.map_preimage MeasureTheory.SimpleFunc.map_preimage +/- warning: measure_theory.simple_func.map_preimage_singleton -> MeasureTheory.SimpleFunc.map_preimage_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : β -> γ) (c : γ), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u3} α γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f)) (Singleton.singleton.{u3, u3} γ (Set.{u3} γ) (Set.hasSingleton.{u3} γ) c)) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) (Finset.filter.{u2} β (fun (b : β) => Eq.{succ u3} γ (g b) c) (fun (a : β) => Classical.propDecidable ((fun (b : β) => Eq.{succ u3} γ (g b) c) a)) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f)))) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : β -> γ) (c : γ), Eq.{succ u3} (Set.{u3} α) (Set.preimage.{u3, u1} α γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.map.{u3, u2, u1} α β γ _inst_1 g f)) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) c)) (Set.preimage.{u3, u2} α β (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f) (Finset.toSet.{u2} β (Finset.filter.{u2} β (fun (b : β) => Eq.{succ u1} γ (g b) c) (fun (a : β) => Classical.propDecidable ((fun (b : β) => Eq.{succ u1} γ (g b) c) a)) (MeasureTheory.SimpleFunc.range.{u3, u2} α β _inst_1 f)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_preimage_singleton MeasureTheory.SimpleFunc.map_preimage_singletonₓ'. -/ theorem map_preimage_singleton (f : α →ₛ β) (g : β → γ) (c : γ) : f.map g ⁻¹' {c} = f ⁻¹' ↑(f.range.filterₓ fun b => g b = c) := map_preimage _ _ _ #align measure_theory.simple_func.map_preimage_singleton MeasureTheory.SimpleFunc.map_preimage_singleton +#print MeasureTheory.SimpleFunc.comp /- /-- Composition of a `simple_fun` and a measurable function is a `simple_func`. -/ def comp [MeasurableSpace β] (f : β →ₛ γ) (g : α → β) (hgm : Measurable g) : α →ₛ γ where @@ -345,18 +588,32 @@ def comp [MeasurableSpace β] (f : β →ₛ γ) (g : α → β) (hgm : Measurab finite_range' := f.finite_range.Subset <| Set.range_comp_subset_range _ _ measurableSet_fiber' z := hgm (f.measurableSet_fiber z) #align measure_theory.simple_func.comp MeasureTheory.SimpleFunc.comp +-/ +/- warning: measure_theory.simple_func.coe_comp -> MeasureTheory.SimpleFunc.coe_comp is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) {g : α -> β} (hgm : Measurable.{u1, u2} α β _inst_1 _inst_2 g), Eq.{max (succ u1) (succ u3)} (α -> γ) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.comp.{u1, u2, u3} α β γ _inst_1 _inst_2 f g hgm)) (Function.comp.{succ u1, succ u2, succ u3} α β γ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (fun (_x : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) => β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u3} β γ _inst_2) f) g) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u3} β] (f : MeasureTheory.SimpleFunc.{u3, u2} β _inst_2 γ) {g : α -> β} (hgm : Measurable.{u1, u3} α β _inst_1 _inst_2 g), Eq.{max (succ u1) (succ u2)} (α -> γ) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 γ (MeasureTheory.SimpleFunc.comp.{u1, u3, u2} α β γ _inst_1 _inst_2 f g hgm)) (Function.comp.{succ u1, succ u3, succ u2} α β γ (MeasureTheory.SimpleFunc.toFun.{u3, u2} β _inst_2 γ f) g) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_comp MeasureTheory.SimpleFunc.coe_compₓ'. -/ @[simp] theorem coe_comp [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) : ⇑(f.comp g hgm) = f ∘ g := rfl #align measure_theory.simple_func.coe_comp MeasureTheory.SimpleFunc.coe_comp +/- warning: measure_theory.simple_func.range_comp_subset_range -> MeasureTheory.SimpleFunc.range_comp_subset_range is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) {g : α -> β} (hgm : Measurable.{u1, u2} α β _inst_1 _inst_2 g), HasSubset.Subset.{u3} (Finset.{u3} γ) (Finset.hasSubset.{u3} γ) (MeasureTheory.SimpleFunc.range.{u1, u3} α γ _inst_1 (MeasureTheory.SimpleFunc.comp.{u1, u2, u3} α β γ _inst_1 _inst_2 f g hgm)) (MeasureTheory.SimpleFunc.range.{u2, u3} β γ _inst_2 f) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u3} β] (f : MeasureTheory.SimpleFunc.{u3, u2} β _inst_2 γ) {g : α -> β} (hgm : Measurable.{u1, u3} α β _inst_1 _inst_2 g), HasSubset.Subset.{u2} (Finset.{u2} γ) (Finset.instHasSubsetFinset.{u2} γ) (MeasureTheory.SimpleFunc.range.{u1, u2} α γ _inst_1 (MeasureTheory.SimpleFunc.comp.{u1, u3, u2} α β γ _inst_1 _inst_2 f g hgm)) (MeasureTheory.SimpleFunc.range.{u3, u2} β γ _inst_2 f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.range_comp_subset_range MeasureTheory.SimpleFunc.range_comp_subset_rangeₓ'. -/ theorem range_comp_subset_range [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) : (f.comp g hgm).range ⊆ f.range := Finset.coe_subset.1 <| by simp only [coe_range, coe_comp, Set.range_comp_subset_range] #align measure_theory.simple_func.range_comp_subset_range MeasureTheory.SimpleFunc.range_comp_subset_range +#print MeasureTheory.SimpleFunc.extend /- /-- Extend a `simple_func` along a measurable embedding: `f₁.extend g hg f₂` is the function `F : β →ₛ γ` such that `F ∘ g = f₁` and `F y = f₂ y` whenever `y ∉ range g`. -/ def extend [MeasurableSpace β] (f₁ : α →ₛ γ) (g : α → β) (hg : MeasurableEmbedding g) @@ -369,59 +626,112 @@ def extend [MeasurableSpace β] (f₁ : α →ₛ γ) (g : α → β) (hg : Meas letI : MeasurableSpace γ := ⊤; haveI : MeasurableSingletonClass γ := ⟨fun _ => trivial⟩ exact fun x => hg.measurable_extend f₁.measurable f₂.measurable (measurable_set_singleton _) #align measure_theory.simple_func.extend MeasureTheory.SimpleFunc.extend +-/ +/- warning: measure_theory.simple_func.extend_apply -> MeasureTheory.SimpleFunc.extend_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f₁ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u1, u2} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (x : α), Eq.{succ u3} γ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (fun (_x : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) => β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u3} β γ _inst_2) (MeasureTheory.SimpleFunc.extend.{u1, u2, u3} α β γ _inst_1 _inst_2 f₁ g hg f₂) (g x)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) f₁ x) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u3} β] (f₁ : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u2, u3} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u3, u1} β _inst_2 γ) (x : α), Eq.{succ u1} γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} β _inst_2 γ (MeasureTheory.SimpleFunc.extend.{u2, u3, u1} α β γ _inst_1 _inst_2 f₁ g hg f₂) (g x)) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 γ f₁ x) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.extend_apply MeasureTheory.SimpleFunc.extend_applyₓ'. -/ @[simp] theorem extend_apply [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) (x : α) : (f₁.extend g hg f₂) (g x) = f₁ x := hg.Injective.extend_apply _ _ _ #align measure_theory.simple_func.extend_apply MeasureTheory.SimpleFunc.extend_apply +/- warning: measure_theory.simple_func.extend_apply' -> MeasureTheory.SimpleFunc.extend_apply' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f₁ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u1, u2} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) {y : β}, (Not (Exists.{succ u1} α (fun (x : α) => Eq.{succ u2} β (g x) y))) -> (Eq.{succ u3} γ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (fun (_x : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) => β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u3} β γ _inst_2) (MeasureTheory.SimpleFunc.extend.{u1, u2, u3} α β γ _inst_1 _inst_2 f₁ g hg f₂) y) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (fun (_x : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) => β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u3} β γ _inst_2) f₂ y)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u3} β] (f₁ : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u2, u3} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u3, u1} β _inst_2 γ) {y : β}, (Not (Exists.{succ u2} α (fun (x : α) => Eq.{succ u3} β (g x) y))) -> (Eq.{succ u1} γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} β _inst_2 γ (MeasureTheory.SimpleFunc.extend.{u2, u3, u1} α β γ _inst_1 _inst_2 f₁ g hg f₂) y) (MeasureTheory.SimpleFunc.toFun.{u3, u1} β _inst_2 γ f₂ y)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.extend_apply' MeasureTheory.SimpleFunc.extend_apply'ₓ'. -/ @[simp] theorem extend_apply' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) {y : β} (h : ¬∃ x, g x = y) : (f₁.extend g hg f₂) y = f₂ y := Function.extend_apply' _ _ _ h #align measure_theory.simple_func.extend_apply' MeasureTheory.SimpleFunc.extend_apply' +/- warning: measure_theory.simple_func.extend_comp_eq' -> MeasureTheory.SimpleFunc.extend_comp_eq' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f₁ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u1, u2} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) (fun (_x : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ) => β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u3} β γ _inst_2) (MeasureTheory.SimpleFunc.extend.{u1, u2, u3} α β γ _inst_1 _inst_2 f₁ g hg f₂)) g) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) f₁) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u3} β] (f₁ : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u2, u3} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u3, u1} β _inst_2 γ), Eq.{max (succ u2) (succ u1)} (α -> γ) (Function.comp.{succ u2, succ u3, succ u1} α β γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} β _inst_2 γ (MeasureTheory.SimpleFunc.extend.{u2, u3, u1} α β γ _inst_1 _inst_2 f₁ g hg f₂)) g) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 γ f₁) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.extend_comp_eq' MeasureTheory.SimpleFunc.extend_comp_eq'ₓ'. -/ @[simp] theorem extend_comp_eq' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) : f₁.extend g hg f₂ ∘ g = f₁ := funext fun x => extend_apply _ _ _ _ #align measure_theory.simple_func.extend_comp_eq' MeasureTheory.SimpleFunc.extend_comp_eq' +/- warning: measure_theory.simple_func.extend_comp_eq -> MeasureTheory.SimpleFunc.extend_comp_eq is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] (f₁ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u1, u2} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u2, u3} β _inst_2 γ), Eq.{max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.comp.{u1, u2, u3} α β γ _inst_1 _inst_2 (MeasureTheory.SimpleFunc.extend.{u1, u2, u3} α β γ _inst_1 _inst_2 f₁ g hg f₂) g (MeasurableEmbedding.measurable.{u1, u2} α β _inst_1 _inst_2 g hg)) f₁ +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u3} β] (f₁ : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) {g : α -> β} (hg : MeasurableEmbedding.{u2, u3} α β _inst_1 _inst_2 g) (f₂ : MeasureTheory.SimpleFunc.{u3, u1} β _inst_2 γ), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) (MeasureTheory.SimpleFunc.comp.{u2, u3, u1} α β γ _inst_1 _inst_2 (MeasureTheory.SimpleFunc.extend.{u2, u3, u1} α β γ _inst_1 _inst_2 f₁ g hg f₂) g (MeasurableEmbedding.measurable.{u2, u3} α β _inst_1 _inst_2 g hg)) f₁ +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.extend_comp_eq MeasureTheory.SimpleFunc.extend_comp_eqₓ'. -/ @[simp] theorem extend_comp_eq [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) : (f₁.extend g hg f₂).comp g hg.Measurable = f₁ := coe_injective <| extend_comp_eq' _ _ _ #align measure_theory.simple_func.extend_comp_eq MeasureTheory.SimpleFunc.extend_comp_eq +#print MeasureTheory.SimpleFunc.seq /- /-- If `f` is a simple function taking values in `β → γ` and `g` is another simple function with the same domain and codomain `β`, then `f.seq g = f a (g a)`. -/ def seq (f : α →ₛ β → γ) (g : α →ₛ β) : α →ₛ γ := f.bind fun f => g.map f #align measure_theory.simple_func.seq MeasureTheory.SimpleFunc.seq +-/ +/- warning: measure_theory.simple_func.seq_apply -> MeasureTheory.SimpleFunc.seq_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (β -> γ)) (g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (a : α), Eq.{succ u3} γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) (MeasureTheory.SimpleFunc.seq.{u1, u2, u3} α β γ _inst_1 f g) a) (coeFn.{max (succ u1) (succ (max u2 u3)), max (succ u1) (succ (max u2 u3))} (MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (β -> γ)) (fun (_x : MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (β -> γ)) => α -> β -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, max u2 u3} α (β -> γ) _inst_1) f a (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g a)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, max u2 u1} α _inst_1 (β -> γ)) (g : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (a : α), Eq.{succ u1} γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ (MeasureTheory.SimpleFunc.seq.{u3, u2, u1} α β γ _inst_1 f g) a) (MeasureTheory.SimpleFunc.toFun.{u3, max u2 u1} α _inst_1 (β -> γ) f a (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β g a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.seq_apply MeasureTheory.SimpleFunc.seq_applyₓ'. -/ @[simp] theorem seq_apply (f : α →ₛ β → γ) (g : α →ₛ β) (a : α) : f.seq g a = f a (g a) := rfl #align measure_theory.simple_func.seq_apply MeasureTheory.SimpleFunc.seq_apply +#print MeasureTheory.SimpleFunc.pair /- /-- Combine two simple functions `f : α →ₛ β` and `g : α →ₛ β` into `λ a, (f a, g a)`. -/ def pair (f : α →ₛ β) (g : α →ₛ γ) : α →ₛ β × γ := (f.map Prod.mk).seq g #align measure_theory.simple_func.pair MeasureTheory.SimpleFunc.pair +-/ +/- warning: measure_theory.simple_func.pair_apply -> MeasureTheory.SimpleFunc.pair_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (a : α), Eq.{max (succ u2) (succ u3)} (Prod.{u2, u3} β γ) (coeFn.{max (succ u1) (succ (max u2 u3)), max (succ u1) (succ (max u2 u3))} (MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) (fun (_x : MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) => α -> (Prod.{u2, u3} β γ)) (MeasureTheory.SimpleFunc.instCoeFun.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1) (MeasureTheory.SimpleFunc.pair.{u1, u2, u3} α β γ _inst_1 f g) a) (Prod.mk.{u2, u3} β γ (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f a) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) g a)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u3, u1} α _inst_1 γ) (a : α), Eq.{max (succ u2) (succ u1)} (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.toFun.{u3, max u2 u1} α _inst_1 (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.pair.{u3, u2, u1} α β γ _inst_1 f g) a) (Prod.mk.{u2, u1} β γ (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f a) (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ g a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.pair_apply MeasureTheory.SimpleFunc.pair_applyₓ'. -/ @[simp] theorem pair_apply (f : α →ₛ β) (g : α →ₛ γ) (a) : pair f g a = (f a, g a) := rfl #align measure_theory.simple_func.pair_apply MeasureTheory.SimpleFunc.pair_apply +/- warning: measure_theory.simple_func.pair_preimage -> MeasureTheory.SimpleFunc.pair_preimage is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (s : Set.{u2} β) (t : Set.{u3} γ), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (coeFn.{max (succ u1) (succ (max u2 u3)), max (succ u1) (succ (max u2 u3))} (MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) (fun (_x : MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) => α -> (Prod.{u2, u3} β γ)) (MeasureTheory.SimpleFunc.instCoeFun.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1) (MeasureTheory.SimpleFunc.pair.{u1, u2, u3} α β γ _inst_1 f g)) (Set.prod.{u2, u3} β γ s t)) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) s) (Set.preimage.{u1, u3} α γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) g) t)) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u3, u1} α _inst_1 γ) (s : Set.{u2} β) (t : Set.{u1} γ), Eq.{succ u3} (Set.{u3} α) (Set.preimage.{u3, max u2 u1} α (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.toFun.{u3, max u2 u1} α _inst_1 (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.pair.{u3, u2, u1} α β γ _inst_1 f g)) (Set.prod.{u2, u1} β γ s t)) (Inter.inter.{u3} (Set.{u3} α) (Set.instInterSet.{u3} α) (Set.preimage.{u3, u2} α β (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f) s) (Set.preimage.{u3, u1} α γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ g) t)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.pair_preimage MeasureTheory.SimpleFunc.pair_preimageₓ'. -/ /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/ theorem pair_preimage (f : α →ₛ β) (g : α →ₛ γ) (s : Set β) (t : Set γ) : pair f g ⁻¹' s ×ˢ t = f ⁻¹' s ∩ g ⁻¹' t := rfl #align measure_theory.simple_func.pair_preimage MeasureTheory.SimpleFunc.pair_preimage +/- warning: measure_theory.simple_func.pair_preimage_singleton -> MeasureTheory.SimpleFunc.pair_preimage_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (b : β) (c : γ), Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (coeFn.{max (succ u1) (succ (max u2 u3)), max (succ u1) (succ (max u2 u3))} (MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) (fun (_x : MeasureTheory.SimpleFunc.{u1, max u2 u3} α _inst_1 (Prod.{u2, u3} β γ)) => α -> (Prod.{u2, u3} β γ)) (MeasureTheory.SimpleFunc.instCoeFun.{u1, max u2 u3} α (Prod.{u2, u3} β γ) _inst_1) (MeasureTheory.SimpleFunc.pair.{u1, u2, u3} α β γ _inst_1 f g)) (Singleton.singleton.{max u2 u3, max u2 u3} (Prod.{u2, u3} β γ) (Set.{max u2 u3} (Prod.{u2, u3} β γ)) (Set.hasSingleton.{max u2 u3} (Prod.{u2, u3} β γ)) (Prod.mk.{u2, u3} β γ b c))) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) b)) (Set.preimage.{u1, u3} α γ (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u3} α γ _inst_1) g) (Singleton.singleton.{u3, u3} γ (Set.{u3} γ) (Set.hasSingleton.{u3} γ) c))) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u3} α] (f : MeasureTheory.SimpleFunc.{u3, u2} α _inst_1 β) (g : MeasureTheory.SimpleFunc.{u3, u1} α _inst_1 γ) (b : β) (c : γ), Eq.{succ u3} (Set.{u3} α) (Set.preimage.{u3, max u2 u1} α (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.toFun.{u3, max u2 u1} α _inst_1 (Prod.{u2, u1} β γ) (MeasureTheory.SimpleFunc.pair.{u3, u2, u1} α β γ _inst_1 f g)) (Singleton.singleton.{max u1 u2, max u2 u1} (Prod.{u2, u1} β γ) (Set.{max u2 u1} (Prod.{u2, u1} β γ)) (Set.instSingletonSet.{max u2 u1} (Prod.{u2, u1} β γ)) (Prod.mk.{u2, u1} β γ b c))) (Inter.inter.{u3} (Set.{u3} α) (Set.instInterSet.{u3} α) (Set.preimage.{u3, u2} α β (MeasureTheory.SimpleFunc.toFun.{u3, u2} α _inst_1 β f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.instSingletonSet.{u2} β) b)) (Set.preimage.{u3, u1} α γ (MeasureTheory.SimpleFunc.toFun.{u3, u1} α _inst_1 γ g) (Singleton.singleton.{u1, u1} γ (Set.{u1} γ) (Set.instSingletonSet.{u1} γ) c))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.pair_preimage_singleton MeasureTheory.SimpleFunc.pair_preimage_singletonₓ'. -/ -- A special form of `pair_preimage` theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) (c : γ) : pair f g ⁻¹' {(b, c)} = f ⁻¹' {b} ∩ g ⁻¹' {c} := @@ -430,6 +740,12 @@ theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) ( exact pair_preimage _ _ _ _ #align measure_theory.simple_func.pair_preimage_singleton MeasureTheory.SimpleFunc.pair_preimage_singleton +/- warning: measure_theory.simple_func.bind_const -> MeasureTheory.SimpleFunc.bind_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.bind.{u1, u2, u2} α β β _inst_1 f (MeasureTheory.SimpleFunc.const.{u1, u2} α β _inst_1)) f +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.bind.{u2, u1, u1} α β β _inst_1 f (MeasureTheory.SimpleFunc.const.{u2, u1} α β _inst_1)) f +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.bind_const MeasureTheory.SimpleFunc.bind_constₓ'. -/ theorem bind_const (f : α →ₛ β) : f.bind (const α) = f := by ext <;> simp #align measure_theory.simple_func.bind_const MeasureTheory.SimpleFunc.bind_const @@ -458,83 +774,120 @@ instance [Inf β] : Inf (α →ₛ β) := instance [LE β] : LE (α →ₛ β) := ⟨fun f g => ∀ a, f a ≤ g a⟩ +#print MeasureTheory.SimpleFunc.const_one /- @[simp, to_additive] theorem const_one [One β] : const α (1 : β) = 1 := rfl #align measure_theory.simple_func.const_one MeasureTheory.SimpleFunc.const_one #align measure_theory.simple_func.const_zero MeasureTheory.SimpleFunc.const_zero +-/ +#print MeasureTheory.SimpleFunc.coe_one /- @[simp, norm_cast, to_additive] theorem coe_one [One β] : ⇑(1 : α →ₛ β) = 1 := rfl #align measure_theory.simple_func.coe_one MeasureTheory.SimpleFunc.coe_one #align measure_theory.simple_func.coe_zero MeasureTheory.SimpleFunc.coe_zero +-/ +#print MeasureTheory.SimpleFunc.coe_mul /- @[simp, norm_cast, to_additive] theorem coe_mul [Mul β] (f g : α →ₛ β) : ⇑(f * g) = f * g := rfl #align measure_theory.simple_func.coe_mul MeasureTheory.SimpleFunc.coe_mul #align measure_theory.simple_func.coe_add MeasureTheory.SimpleFunc.coe_add +-/ +#print MeasureTheory.SimpleFunc.coe_inv /- @[simp, norm_cast, to_additive] theorem coe_inv [Inv β] (f : α →ₛ β) : ⇑f⁻¹ = f⁻¹ := rfl #align measure_theory.simple_func.coe_inv MeasureTheory.SimpleFunc.coe_inv #align measure_theory.simple_func.coe_neg MeasureTheory.SimpleFunc.coe_neg +-/ +#print MeasureTheory.SimpleFunc.coe_div /- @[simp, norm_cast, to_additive] theorem coe_div [Div β] (f g : α →ₛ β) : ⇑(f / g) = f / g := rfl #align measure_theory.simple_func.coe_div MeasureTheory.SimpleFunc.coe_div #align measure_theory.simple_func.coe_sub MeasureTheory.SimpleFunc.coe_sub +-/ +/- warning: measure_theory.simple_func.coe_le -> MeasureTheory.SimpleFunc.coe_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Preorder.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, Iff (LE.le.{max u1 u2} ((fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) f) (Pi.hasLe.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => Preorder.toHasLe.{u2} β _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) g)) (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toHasLe.{u2} β _inst_2)) f g) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Preorder.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, Iff (LE.le.{max u1 u2} (α -> β) (Pi.hasLe.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => Preorder.toLE.{u2} β _inst_2)) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β f) (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β g)) (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toLE.{u2} β _inst_2)) f g) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_le MeasureTheory.SimpleFunc.coe_leₓ'. -/ @[simp, norm_cast] theorem coe_le [Preorder β] {f g : α →ₛ β} : (f : α → β) ≤ g ↔ f ≤ g := Iff.rfl #align measure_theory.simple_func.coe_le MeasureTheory.SimpleFunc.coe_le +#print MeasureTheory.SimpleFunc.coe_sup /- @[simp, norm_cast] theorem coe_sup [Sup β] (f g : α →ₛ β) : ⇑(f ⊔ g) = f ⊔ g := rfl #align measure_theory.simple_func.coe_sup MeasureTheory.SimpleFunc.coe_sup +-/ +#print MeasureTheory.SimpleFunc.coe_inf /- @[simp, norm_cast] theorem coe_inf [Inf β] (f g : α →ₛ β) : ⇑(f ⊓ g) = f ⊓ g := rfl #align measure_theory.simple_func.coe_inf MeasureTheory.SimpleFunc.coe_inf +-/ +#print MeasureTheory.SimpleFunc.mul_apply /- @[to_additive] theorem mul_apply [Mul β] (f g : α →ₛ β) (a : α) : (f * g) a = f a * g a := rfl #align measure_theory.simple_func.mul_apply MeasureTheory.SimpleFunc.mul_apply #align measure_theory.simple_func.add_apply MeasureTheory.SimpleFunc.add_apply +-/ +#print MeasureTheory.SimpleFunc.div_apply /- @[to_additive] theorem div_apply [Div β] (f g : α →ₛ β) (x : α) : (f / g) x = f x / g x := rfl #align measure_theory.simple_func.div_apply MeasureTheory.SimpleFunc.div_apply #align measure_theory.simple_func.sub_apply MeasureTheory.SimpleFunc.sub_apply +-/ +#print MeasureTheory.SimpleFunc.inv_apply /- @[to_additive] theorem inv_apply [Inv β] (f : α →ₛ β) (x : α) : f⁻¹ x = (f x)⁻¹ := rfl #align measure_theory.simple_func.inv_apply MeasureTheory.SimpleFunc.inv_apply #align measure_theory.simple_func.neg_apply MeasureTheory.SimpleFunc.neg_apply +-/ +#print MeasureTheory.SimpleFunc.sup_apply /- theorem sup_apply [Sup β] (f g : α →ₛ β) (a : α) : (f ⊔ g) a = f a ⊔ g a := rfl #align measure_theory.simple_func.sup_apply MeasureTheory.SimpleFunc.sup_apply +-/ +#print MeasureTheory.SimpleFunc.inf_apply /- theorem inf_apply [Inf β] (f g : α →ₛ β) (a : α) : (f ⊓ g) a = f a ⊓ g a := rfl #align measure_theory.simple_func.inf_apply MeasureTheory.SimpleFunc.inf_apply +-/ +/- warning: measure_theory.simple_func.range_one -> MeasureTheory.SimpleFunc.range_one is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Nonempty.{succ u1} α] [_inst_3 : One.{u2} β], Eq.{succ u2} (Finset.{u2} β) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 (OfNat.ofNat.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 1 (OfNat.mk.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 1 (One.one.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instOne.{u1, u2} α β _inst_1 _inst_3))))) (Singleton.singleton.{u2, u2} β (Finset.{u2} β) (Finset.hasSingleton.{u2} β) (OfNat.ofNat.{u2} β 1 (OfNat.mk.{u2} β 1 (One.one.{u2} β _inst_3)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Nonempty.{succ u2} α] [_inst_3 : One.{u1} β], Eq.{succ u1} (Finset.{u1} β) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 (OfNat.ofNat.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) 1 (One.toOfNat1.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.instOne.{u2, u1} α β _inst_1 _inst_3)))) (Singleton.singleton.{u1, u1} β (Finset.{u1} β) (Finset.instSingletonFinset.{u1} β) (OfNat.ofNat.{u1} β 1 (One.toOfNat1.{u1} β _inst_3))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.range_one MeasureTheory.SimpleFunc.range_oneₓ'. -/ @[simp, to_additive] theorem range_one [Nonempty α] [One β] : (1 : α →ₛ β).range = {1} := Finset.ext fun x => by simp [eq_comm] #align measure_theory.simple_func.range_one MeasureTheory.SimpleFunc.range_one #align measure_theory.simple_func.range_zero MeasureTheory.SimpleFunc.range_zero +#print MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty /- @[simp] theorem range_eq_empty_of_isEmpty {β} [hα : IsEmpty α] (f : α →ₛ β) : f.range = ∅ := by @@ -546,27 +899,42 @@ theorem range_eq_empty_of_isEmpty {β} [hα : IsEmpty α] (f : α →ₛ β) : f rw [isEmpty_iff] at hα exact hα x #align measure_theory.simple_func.range_eq_empty_of_is_empty MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty +-/ +#print MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero /- theorem eq_zero_of_mem_range_zero [Zero β] : ∀ {y : β}, y ∈ (0 : α →ₛ β).range → y = 0 := forall_range_iff.2 fun x => rfl #align measure_theory.simple_func.eq_zero_of_mem_range_zero MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero +-/ +#print MeasureTheory.SimpleFunc.mul_eq_map₂ /- @[to_additive] theorem mul_eq_map₂ [Mul β] (f g : α →ₛ β) : f * g = (pair f g).map fun p : β × β => p.1 * p.2 := rfl #align measure_theory.simple_func.mul_eq_map₂ MeasureTheory.SimpleFunc.mul_eq_map₂ #align measure_theory.simple_func.add_eq_map₂ MeasureTheory.SimpleFunc.add_eq_map₂ +-/ +#print MeasureTheory.SimpleFunc.sup_eq_map₂ /- theorem sup_eq_map₂ [Sup β] (f g : α →ₛ β) : f ⊔ g = (pair f g).map fun p : β × β => p.1 ⊔ p.2 := rfl #align measure_theory.simple_func.sup_eq_map₂ MeasureTheory.SimpleFunc.sup_eq_map₂ +-/ +#print MeasureTheory.SimpleFunc.const_mul_eq_map /- @[to_additive] theorem const_mul_eq_map [Mul β] (f : α →ₛ β) (b : β) : const α b * f = f.map fun a => b * a := rfl #align measure_theory.simple_func.const_mul_eq_map MeasureTheory.SimpleFunc.const_mul_eq_map #align measure_theory.simple_func.const_add_eq_map MeasureTheory.SimpleFunc.const_add_eq_map +-/ +/- warning: measure_theory.simple_func.map_mul -> MeasureTheory.SimpleFunc.map_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Mul.{u2} β] [_inst_3 : Mul.{u3} γ] {g : β -> γ}, (forall (x : β) (y : β), Eq.{succ u3} γ (g (HMul.hMul.{u2, u2, u2} β β β (instHMul.{u2} β _inst_2) x y)) (HMul.hMul.{u3, u3, u3} γ γ γ (instHMul.{u3} γ _inst_3) (g x) (g y))) -> (forall (f₁ : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (f₂ : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u3)} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (instHMul.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instMul.{u1, u2} α β _inst_1 _inst_2)) f₁ f₂)) (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (instHMul.{max u1 u3} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 γ) (MeasureTheory.SimpleFunc.instMul.{u1, u3} α γ _inst_1 _inst_3)) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f₁) (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ _inst_1 g f₂))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Mul.{u3} β] [_inst_3 : Mul.{u2} γ] {g : β -> γ}, (forall (x : β) (y : β), Eq.{succ u2} γ (g (HMul.hMul.{u3, u3, u3} β β β (instHMul.{u3} β _inst_2) x y)) (HMul.hMul.{u2, u2, u2} γ γ γ (instHMul.{u2} γ _inst_3) (g x) (g y))) -> (forall (f₁ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β) (f₂ : MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.map.{u1, u3, u2} α β γ _inst_1 g (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β) (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β) (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β) (instHMul.{max u1 u3} (MeasureTheory.SimpleFunc.{u1, u3} α _inst_1 β) (MeasureTheory.SimpleFunc.instMul.{u1, u3} α β _inst_1 _inst_2)) f₁ f₂)) (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (instHMul.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.instMul.{u1, u2} α γ _inst_1 _inst_3)) (MeasureTheory.SimpleFunc.map.{u1, u3, u2} α β γ _inst_1 g f₁) (MeasureTheory.SimpleFunc.map.{u1, u3, u2} α β γ _inst_1 g f₂))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_mul MeasureTheory.SimpleFunc.map_mulₓ'. -/ @[to_additive] theorem map_mul [Mul β] [Mul γ] {g : β → γ} (hg : ∀ x y, g (x * y) = g x * g y) (f₁ f₂ : α →ₛ β) : (f₁ * f₂).map g = f₁.map g * f₂.map g := @@ -579,40 +947,56 @@ variable {K : Type _} instance [SMul K β] : SMul K (α →ₛ β) := ⟨fun k f => f.map ((· • ·) k)⟩ +#print MeasureTheory.SimpleFunc.coe_smul /- @[simp] theorem coe_smul [SMul K β] (c : K) (f : α →ₛ β) : ⇑(c • f) = c • f := rfl #align measure_theory.simple_func.coe_smul MeasureTheory.SimpleFunc.coe_smul +-/ +#print MeasureTheory.SimpleFunc.smul_apply /- theorem smul_apply [SMul K β] (k : K) (f : α →ₛ β) (a : α) : (k • f) a = k • f a := rfl #align measure_theory.simple_func.smul_apply MeasureTheory.SimpleFunc.smul_apply +-/ +#print MeasureTheory.SimpleFunc.hasNatPow /- instance hasNatPow [Monoid β] : Pow (α →ₛ β) ℕ := ⟨fun f n => f.map (· ^ n)⟩ #align measure_theory.simple_func.has_nat_pow MeasureTheory.SimpleFunc.hasNatPow +-/ +#print MeasureTheory.SimpleFunc.coe_pow /- @[simp] theorem coe_pow [Monoid β] (f : α →ₛ β) (n : ℕ) : ⇑(f ^ n) = f ^ n := rfl #align measure_theory.simple_func.coe_pow MeasureTheory.SimpleFunc.coe_pow +-/ +#print MeasureTheory.SimpleFunc.pow_apply /- theorem pow_apply [Monoid β] (n : ℕ) (f : α →ₛ β) (a : α) : (f ^ n) a = f a ^ n := rfl #align measure_theory.simple_func.pow_apply MeasureTheory.SimpleFunc.pow_apply +-/ +#print MeasureTheory.SimpleFunc.hasIntPow /- instance hasIntPow [DivInvMonoid β] : Pow (α →ₛ β) ℤ := ⟨fun f n => f.map (· ^ n)⟩ #align measure_theory.simple_func.has_int_pow MeasureTheory.SimpleFunc.hasIntPow +-/ +#print MeasureTheory.SimpleFunc.coe_zpow /- @[simp] theorem coe_zpow [DivInvMonoid β] (f : α →ₛ β) (z : ℤ) : ⇑(f ^ z) = f ^ z := rfl #align measure_theory.simple_func.coe_zpow MeasureTheory.SimpleFunc.coe_zpow +-/ +#print MeasureTheory.SimpleFunc.zpow_apply /- theorem zpow_apply [DivInvMonoid β] (z : ℤ) (f : α →ₛ β) (a : α) : (f ^ z) a = f a ^ z := rfl #align measure_theory.simple_func.zpow_apply MeasureTheory.SimpleFunc.zpow_apply +-/ -- TODO: work out how to generate these instances with `to_additive`, which gets confused by the -- argument order swap between `coe_smul` and `coe_pow`. @@ -657,17 +1041,19 @@ instance [CommGroup β] : CommGroup (α →ₛ β) := instance [Semiring K] [AddCommMonoid β] [Module K β] : Module K (α →ₛ β) := Function.Injective.module K ⟨fun f => show α → β from f, coe_zero, coe_add⟩ coe_injective coe_smul +#print MeasureTheory.SimpleFunc.smul_eq_map /- theorem smul_eq_map [SMul K β] (k : K) (f : α →ₛ β) : k • f = f.map ((· • ·) k) := rfl #align measure_theory.simple_func.smul_eq_map MeasureTheory.SimpleFunc.smul_eq_map +-/ instance [Preorder β] : Preorder (α →ₛ β) := - { SimpleFunc.hasLe with + { SimpleFunc.instLE with le_refl := fun f a => le_rfl le_trans := fun f g h hfg hgh a => le_trans (hfg _) (hgh a) } instance [PartialOrder β] : PartialOrder (α →ₛ β) := - { SimpleFunc.preorder with + { SimpleFunc.instPreorder with le_antisymm := fun f g hfg hgf => ext fun a => le_antisymm (hfg a) (hgf a) } instance [LE β] [OrderBot β] : OrderBot (α →ₛ β) @@ -681,25 +1067,31 @@ instance [LE β] [OrderTop β] : OrderTop (α →ₛ β) le_top f a := le_top instance [SemilatticeInf β] : SemilatticeInf (α →ₛ β) := - { SimpleFunc.partialOrder with + { SimpleFunc.instPartialOrder with inf := (· ⊓ ·) inf_le_left := fun f g a => inf_le_left inf_le_right := fun f g a => inf_le_right le_inf := fun f g h hfh hgh a => le_inf (hfh a) (hgh a) } instance [SemilatticeSup β] : SemilatticeSup (α →ₛ β) := - { SimpleFunc.partialOrder with + { SimpleFunc.instPartialOrder with sup := (· ⊔ ·) le_sup_left := fun f g a => le_sup_left le_sup_right := fun f g a => le_sup_right sup_le := fun f g h hfh hgh a => sup_le (hfh a) (hgh a) } instance [Lattice β] : Lattice (α →ₛ β) := - { SimpleFunc.semilatticeSup, SimpleFunc.semilatticeInf with } + { SimpleFunc.instSemilatticeSup, SimpleFunc.instSemilatticeInf with } instance [LE β] [BoundedOrder β] : BoundedOrder (α →ₛ β) := - { SimpleFunc.orderBot, SimpleFunc.orderTop with } - + { SimpleFunc.instOrderBot, SimpleFunc.instOrderTop with } + +/- warning: measure_theory.simple_func.finset_sup_apply -> MeasureTheory.SimpleFunc.finset_sup_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] {f : γ -> (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β)} (s : Finset.{u3} γ) (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (Finset.sup.{max u1 u2, u3} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) γ (MeasureTheory.SimpleFunc.instSemilatticeSup.{u1, u2} α β _inst_1 _inst_2) (MeasureTheory.SimpleFunc.instOrderBot.{u1, u2} α β _inst_1 (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) _inst_3) s f) a) (Finset.sup.{u2, u3} β γ _inst_2 _inst_3 s (fun (c : γ) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (f c) a)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : SemilatticeSup.{u3} β] [_inst_3 : OrderBot.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_2)))] {f : γ -> (MeasureTheory.SimpleFunc.{u2, u3} α _inst_1 β)} (s : Finset.{u1} γ) (a : α), Eq.{succ u3} β (MeasureTheory.SimpleFunc.toFun.{u2, u3} α _inst_1 β (Finset.sup.{max u2 u3, u1} (MeasureTheory.SimpleFunc.{u2, u3} α _inst_1 β) γ (MeasureTheory.SimpleFunc.instSemilatticeSup.{u2, u3} α β _inst_1 _inst_2) (MeasureTheory.SimpleFunc.instOrderBot.{u2, u3} α β _inst_1 (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_2))) _inst_3) s f) a) (Finset.sup.{u3, u1} β γ _inst_2 _inst_3 s (fun (c : γ) => MeasureTheory.SimpleFunc.toFun.{u2, u3} α _inst_1 β (f c) a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.finset_sup_apply MeasureTheory.SimpleFunc.finset_sup_applyₓ'. -/ theorem finset_sup_apply [SemilatticeSup β] [OrderBot β] {f : γ → α →ₛ β} (s : Finset γ) (a : α) : s.sup f a = s.sup fun c => f c a := by @@ -712,17 +1104,31 @@ section Restrict variable [Zero β] +#print MeasureTheory.SimpleFunc.restrict /- /-- Restrict a simple function `f : α →ₛ β` to a set `s`. If `s` is measurable, then `f.restrict s a = if a ∈ s then f a else 0`, otherwise `f.restrict s = const α 0`. -/ def restrict (f : α →ₛ β) (s : Set α) : α →ₛ β := if hs : MeasurableSet s then piecewise s hs f 0 else 0 #align measure_theory.simple_func.restrict MeasureTheory.SimpleFunc.restrict +-/ +/- warning: measure_theory.simple_func.restrict_of_not_measurable -> MeasureTheory.SimpleFunc.restrict_of_not_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {s : Set.{u1} α}, (Not (MeasurableSet.{u1} α _inst_1 s)) -> (Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s) (OfNat.ofNat.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 0 (OfNat.mk.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 0 (Zero.zero.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instZero.{u1, u2} α β _inst_1 _inst_2))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β} {s : Set.{u2} α}, (Not (MeasurableSet.{u2} α _inst_1 s)) -> (Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s) (OfNat.ofNat.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) 0 (Zero.toOfNat0.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.instZero.{u2, u1} α β _inst_1 _inst_2)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_of_not_measurable MeasureTheory.SimpleFunc.restrict_of_not_measurableₓ'. -/ theorem restrict_of_not_measurable {f : α →ₛ β} {s : Set α} (hs : ¬MeasurableSet s) : restrict f s = 0 := dif_neg hs #align measure_theory.simple_func.restrict_of_not_measurable MeasureTheory.SimpleFunc.restrict_of_not_measurable +/- warning: measure_theory.simple_func.coe_restrict -> MeasureTheory.SimpleFunc.coe_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (Eq.{max (succ u1) (succ u2)} (α -> β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s)) (Set.indicator.{u1, u2} α β _inst_2 s (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) {s : Set.{u2} α}, (MeasurableSet.{u2} α _inst_1 s) -> (Eq.{max (succ u2) (succ u1)} (α -> β) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s)) (Set.indicator.{u2, u1} α β _inst_2 s (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.coe_restrict MeasureTheory.SimpleFunc.coe_restrictₓ'. -/ @[simp] theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) : ⇑(restrict f s) = indicator s f := @@ -731,50 +1137,106 @@ theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) : rfl #align measure_theory.simple_func.coe_restrict MeasureTheory.SimpleFunc.coe_restrict +/- warning: measure_theory.simple_func.restrict_univ -> MeasureTheory.SimpleFunc.restrict_univ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f (Set.univ.{u1} α)) f +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f (Set.univ.{u2} α)) f +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_univ MeasureTheory.SimpleFunc.restrict_univₓ'. -/ @[simp] theorem restrict_univ (f : α →ₛ β) : restrict f univ = f := by simp [restrict] #align measure_theory.simple_func.restrict_univ MeasureTheory.SimpleFunc.restrict_univ +/- warning: measure_theory.simple_func.restrict_empty -> MeasureTheory.SimpleFunc.restrict_empty is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (OfNat.ofNat.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 0 (OfNat.mk.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) 0 (Zero.zero.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instZero.{u1, u2} α β _inst_1 _inst_2)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f (EmptyCollection.emptyCollection.{u2} (Set.{u2} α) (Set.instEmptyCollectionSet.{u2} α))) (OfNat.ofNat.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) 0 (Zero.toOfNat0.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) (MeasureTheory.SimpleFunc.instZero.{u2, u1} α β _inst_1 _inst_2))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_empty MeasureTheory.SimpleFunc.restrict_emptyₓ'. -/ @[simp] theorem restrict_empty (f : α →ₛ β) : restrict f ∅ = 0 := by simp [restrict] #align measure_theory.simple_func.restrict_empty MeasureTheory.SimpleFunc.restrict_empty +#print MeasureTheory.SimpleFunc.map_restrict_of_zero /- theorem map_restrict_of_zero [Zero γ] {g : β → γ} (hg : g 0 = 0) (f : α →ₛ β) (s : Set α) : (f.restrict s).map g = (f.map g).restrict s := ext fun x => if hs : MeasurableSet s then by simp [hs, Set.indicator_comp_of_zero hg] else by simp [restrict_of_not_measurable hs, hg] #align measure_theory.simple_func.map_restrict_of_zero MeasureTheory.SimpleFunc.map_restrict_of_zero +-/ -theorem map_coe_eNNReal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : +/- warning: measure_theory.simple_func.map_coe_ennreal_restrict -> MeasureTheory.SimpleFunc.map_coe_ennreal_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (s : Set.{u1} α), Eq.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal _inst_1 ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe)))) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α NNReal _inst_1 (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))) f s)) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal _inst_1 ENNReal.hasZero (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal _inst_1 ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe)))) f) s) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (s : Set.{u1} α), Eq.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal _inst_1 ENNReal.some (MeasureTheory.SimpleFunc.restrict.{u1, 0} α NNReal _inst_1 instNNRealZero f s)) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal _inst_1 instENNRealZero (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal _inst_1 ENNReal.some f) s) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_coe_ennreal_restrict MeasureTheory.SimpleFunc.map_coe_ennreal_restrictₓ'. -/ +theorem map_coe_ennreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : (f.restrict s).map (coe : ℝ≥0 → ℝ≥0∞) = (f.map coe).restrict s := map_restrict_of_zero ENNReal.coe_zero _ _ -#align measure_theory.simple_func.map_coe_ennreal_restrict MeasureTheory.SimpleFunc.map_coe_eNNReal_restrict - -theorem map_coe_nNReal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : +#align measure_theory.simple_func.map_coe_ennreal_restrict MeasureTheory.SimpleFunc.map_coe_ennreal_restrict + +/- warning: measure_theory.simple_func.map_coe_nnreal_restrict -> MeasureTheory.SimpleFunc.map_coe_nnreal_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (s : Set.{u1} α), Eq.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Real) (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal Real _inst_1 ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe)))) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α NNReal _inst_1 (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))) f s)) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α Real _inst_1 Real.hasZero (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal Real _inst_1 ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal Real (HasLiftT.mk.{1, 1} NNReal Real (CoeTCₓ.coe.{1, 1} NNReal Real (coeBase.{1, 1} NNReal Real NNReal.Real.hasCoe)))) f) s) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (s : Set.{u1} α), Eq.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Real) (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal Real _inst_1 NNReal.toReal (MeasureTheory.SimpleFunc.restrict.{u1, 0} α NNReal _inst_1 instNNRealZero f s)) (MeasureTheory.SimpleFunc.restrict.{u1, 0} α Real _inst_1 Real.instZeroReal (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal Real _inst_1 NNReal.toReal f) s) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_coe_nnreal_restrict MeasureTheory.SimpleFunc.map_coe_nnreal_restrictₓ'. -/ +theorem map_coe_nnreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : (f.restrict s).map (coe : ℝ≥0 → ℝ) = (f.map coe).restrict s := map_restrict_of_zero NNReal.coe_zero _ _ -#align measure_theory.simple_func.map_coe_nnreal_restrict MeasureTheory.SimpleFunc.map_coe_nNReal_restrict - +#align measure_theory.simple_func.map_coe_nnreal_restrict MeasureTheory.SimpleFunc.map_coe_nnreal_restrict + +/- warning: measure_theory.simple_func.restrict_apply -> MeasureTheory.SimpleFunc.restrict_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (forall (a : α), Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s) a) (Set.indicator.{u1, u2} α β _inst_2 s (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) a)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) {s : Set.{u2} α}, (MeasurableSet.{u2} α _inst_1 s) -> (forall (a : α), Eq.{succ u1} β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s) a) (Set.indicator.{u2, u1} α β _inst_2 s (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_apply MeasureTheory.SimpleFunc.restrict_applyₓ'. -/ theorem restrict_apply (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) (a) : restrict f s a = indicator s f a := by simp only [f.coe_restrict hs] #align measure_theory.simple_func.restrict_apply MeasureTheory.SimpleFunc.restrict_apply +/- warning: measure_theory.simple_func.restrict_preimage -> MeasureTheory.SimpleFunc.restrict_preimage is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (forall {t : Set.{u2} β}, (Not (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2))) t)) -> (Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s)) t) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) t)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) {s : Set.{u2} α}, (MeasurableSet.{u2} α _inst_1 s) -> (forall {t : Set.{u1} β}, (Not (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2)) t)) -> (Eq.{succ u2} (Set.{u2} α) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s)) t) (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) s (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) t)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_preimage MeasureTheory.SimpleFunc.restrict_preimageₓ'. -/ theorem restrict_preimage (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : (0 : β) ∉ t) : restrict f s ⁻¹' t = s ∩ f ⁻¹' t := by simp [hs, indicator_preimage_of_not_mem _ _ ht, inter_comm] #align measure_theory.simple_func.restrict_preimage MeasureTheory.SimpleFunc.restrict_preimage +/- warning: measure_theory.simple_func.restrict_preimage_singleton -> MeasureTheory.SimpleFunc.restrict_preimage_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (forall {r : β}, (Ne.{succ u2} β r (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) -> (Eq.{succ u1} (Set.{u1} α) (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s)) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) r)) (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) r))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β) {s : Set.{u2} α}, (MeasurableSet.{u2} α _inst_1 s) -> (forall {r : β}, (Ne.{succ u1} β r (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) -> (Eq.{succ u2} (Set.{u2} α) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s)) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) r)) (Inter.inter.{u2} (Set.{u2} α) (Set.instInterSet.{u2} α) s (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) r))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_preimage_singleton MeasureTheory.SimpleFunc.restrict_preimage_singletonₓ'. -/ theorem restrict_preimage_singleton (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {r : β} (hr : r ≠ 0) : restrict f s ⁻¹' {r} = s ∩ f ⁻¹' {r} := f.restrictPreimage hs hr.symm #align measure_theory.simple_func.restrict_preimage_singleton MeasureTheory.SimpleFunc.restrict_preimage_singleton +/- warning: measure_theory.simple_func.mem_restrict_range -> MeasureTheory.SimpleFunc.mem_restrict_range is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] {r : β} {s : Set.{u1} α} {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, (MeasurableSet.{u1} α _inst_1 s) -> (Iff (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) r (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s))) (Or (And (Eq.{succ u2} β r (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) (Ne.{succ u1} (Set.{u1} α) s (Set.univ.{u1} α))) (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) r (Set.image.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) s)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] {r : β} {s : Set.{u2} α} {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β}, (MeasurableSet.{u2} α _inst_1 s) -> (Iff (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) r (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s))) (Or (And (Eq.{succ u1} β r (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) (Ne.{succ u2} (Set.{u2} α) s (Set.univ.{u2} α))) (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) r (Set.image.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) s)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.mem_restrict_range MeasureTheory.SimpleFunc.mem_restrict_rangeₓ'. -/ theorem mem_restrict_range {r : β} {s : Set α} {f : α →ₛ β} (hs : MeasurableSet s) : r ∈ (restrict f s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ f '' s := by rw [← Finset.mem_coe, coe_range, coe_restrict _ hs, mem_range_indicator] #align measure_theory.simple_func.mem_restrict_range MeasureTheory.SimpleFunc.mem_restrict_range +/- warning: measure_theory.simple_func.mem_image_of_mem_range_restrict -> MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] {r : β} {s : Set.{u1} α} {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) r (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s))) -> (Ne.{succ u2} β r (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) r (Set.image.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) s)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] {r : β} {s : Set.{u2} α} {f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β}, (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) r (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 (MeasureTheory.SimpleFunc.restrict.{u2, u1} α β _inst_1 _inst_2 f s))) -> (Ne.{succ u1} β r (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) -> (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) r (Set.image.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) s)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.mem_image_of_mem_range_restrict MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrictₓ'. -/ theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β} (hr : r ∈ (restrict f s).range) (h0 : r ≠ 0) : r ∈ f '' s := if hs : MeasurableSet s then by simpa [mem_restrict_range hs, h0] using hr @@ -783,6 +1245,12 @@ theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β} exact (h0 <| eq_zero_of_mem_range_zero hr).elim #align measure_theory.simple_func.mem_image_of_mem_range_restrict MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict +/- warning: measure_theory.simple_func.restrict_mono -> MeasureTheory.SimpleFunc.restrict_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] [_inst_3 : Preorder.{u2} β] (s : Set.{u1} α) {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toHasLe.{u2} β _inst_3)) f g) -> (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toHasLe.{u2} β _inst_3)) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 g s)) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] [_inst_3 : Preorder.{u2} β] (s : Set.{u1} α) {f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β} {g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β}, (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toLE.{u2} β _inst_3)) f g) -> (LE.le.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (MeasureTheory.SimpleFunc.instLE.{u1, u2} α β _inst_1 (Preorder.toLE.{u2} β _inst_3)) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 f s) (MeasureTheory.SimpleFunc.restrict.{u1, u2} α β _inst_1 _inst_2 g s)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_mono MeasureTheory.SimpleFunc.restrict_monoₓ'. -/ @[mono] theorem restrict_mono [Preorder β] (s : Set α) {f g : α →ₛ β} (H : f ≤ g) : f.restrict s ≤ g.restrict s := @@ -799,6 +1267,12 @@ section variable [SemilatticeSup β] [OrderBot β] [Zero β] +/- warning: measure_theory.simple_func.approx -> MeasureTheory.SimpleFunc.approx is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β], (Nat -> β) -> (α -> β) -> Nat -> (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β], (Nat -> β) -> (α -> β) -> Nat -> (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.approx MeasureTheory.SimpleFunc.approxₓ'. -/ /-- Fix a sequence `i : ℕ → β`. Given a function `α → β`, its `n`-th approximation by simple functions is defined so that in case `β = ℝ≥0∞` it sends each `a` to the supremum of the set `{i k | k ≤ n ∧ i k ≤ f a}`, see `approx_apply` and `supr_approx_apply` for details. -/ @@ -806,6 +1280,12 @@ def approx (i : ℕ → β) (f : α → β) (n : ℕ) : α →ₛ β := (Finset.range n).sup fun k => restrict (const α (i k)) { a : α | i k ≤ f a } #align measure_theory.simple_func.approx MeasureTheory.SimpleFunc.approx +/- warning: measure_theory.simple_func.approx_apply -> MeasureTheory.SimpleFunc.approx_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : OrderClosedTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))] [_inst_7 : MeasurableSpace.{u2} β] [_inst_8 : OpensMeasurableSpace.{u2} β _inst_5 _inst_7] {i : Nat -> β} {f : α -> β} {n : Nat} (a : α), (Measurable.{u1, u2} α β _inst_1 _inst_7 f) -> (Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 i f n) a) (Finset.sup.{u2, 0} β Nat _inst_2 _inst_3 (Finset.range n) (fun (k : Nat) => ite.{succ u2} β (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (i k) (f a)) (Classical.propDecidable (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (i k) (f a))) (i k) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_4)))))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : OrderClosedTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))] [_inst_7 : MeasurableSpace.{u2} β] [_inst_8 : OpensMeasurableSpace.{u2} β _inst_5 _inst_7] {i : Nat -> β} {f : α -> β} {n : Nat} (a : α), (Measurable.{u1, u2} α β _inst_1 _inst_7 f) -> (Eq.{succ u2} β (MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 i f n) a) (Finset.sup.{u2, 0} β Nat _inst_2 _inst_3 (Finset.range n) (fun (k : Nat) => ite.{succ u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (i k) (f a)) (Classical.propDecidable (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (i k) (f a))) (i k) (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_4))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.approx_apply MeasureTheory.SimpleFunc.approx_applyₓ'. -/ theorem approx_apply [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : Measurable f) : (approx i f n : α →ₛ β) a = (Finset.range n).sup fun k => if i k ≤ f a then i k else 0 := @@ -819,10 +1299,22 @@ theorem approx_apply [TopologicalSpace β] [OrderClosedTopology β] [MeasurableS exact hf measurableSet_Ici #align measure_theory.simple_func.approx_apply MeasureTheory.SimpleFunc.approx_apply +/- warning: measure_theory.simple_func.monotone_approx -> MeasureTheory.SimpleFunc.monotone_approx is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β] (i : Nat -> β) (f : α -> β), Monotone.{0, max u1 u2} Nat (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (MeasureTheory.SimpleFunc.instPreorder.{u1, u2} α β _inst_1 (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 i f) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β] (i : Nat -> β) (f : α -> β), Monotone.{0, max u2 u1} Nat (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (MeasureTheory.SimpleFunc.instPreorder.{u1, u2} α β _inst_1 (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))) (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 i f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.monotone_approx MeasureTheory.SimpleFunc.monotone_approxₓ'. -/ theorem monotone_approx (i : ℕ → β) (f : α → β) : Monotone (approx i f) := fun n m h => Finset.sup_mono <| Finset.range_subset.2 h #align measure_theory.simple_func.monotone_approx MeasureTheory.SimpleFunc.monotone_approx +/- warning: measure_theory.simple_func.approx_comp -> MeasureTheory.SimpleFunc.approx_comp is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u2} β] [_inst_3 : OrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2)))] [_inst_4 : Zero.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : OrderClosedTopology.{u2} β _inst_5 (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_2))] [_inst_7 : MeasurableSpace.{u2} β] [_inst_8 : OpensMeasurableSpace.{u2} β _inst_5 _inst_7] [_inst_9 : MeasurableSpace.{u3} γ] {i : Nat -> β} {f : γ -> β} {g : α -> γ} {n : Nat} (a : α), (Measurable.{u3, u2} γ β _inst_9 _inst_7 f) -> (Measurable.{u1, u3} α γ _inst_1 _inst_9 g) -> (Eq.{succ u2} β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 i (Function.comp.{succ u1, succ u3, succ u2} α γ β f g) n) a) (coeFn.{max (succ u3) (succ u2), max (succ u3) (succ u2)} (MeasureTheory.SimpleFunc.{u3, u2} γ _inst_9 β) (fun (_x : MeasureTheory.SimpleFunc.{u3, u2} γ _inst_9 β) => γ -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u3, u2} γ β _inst_9) (MeasureTheory.SimpleFunc.approx.{u3, u2} γ β _inst_9 _inst_2 _inst_3 _inst_4 i f n) (g a))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : SemilatticeSup.{u3} β] [_inst_3 : OrderBot.{u3} β (Preorder.toLE.{u3} β (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_2)))] [_inst_4 : Zero.{u3} β] [_inst_5 : TopologicalSpace.{u3} β] [_inst_6 : OrderClosedTopology.{u3} β _inst_5 (PartialOrder.toPreorder.{u3} β (SemilatticeSup.toPartialOrder.{u3} β _inst_2))] [_inst_7 : MeasurableSpace.{u3} β] [_inst_8 : OpensMeasurableSpace.{u3} β _inst_5 _inst_7] [_inst_9 : MeasurableSpace.{u2} γ] {i : Nat -> β} {f : γ -> β} {g : α -> γ} {n : Nat} (a : α), (Measurable.{u2, u3} γ β _inst_9 _inst_7 f) -> (Measurable.{u1, u2} α γ _inst_1 _inst_9 g) -> (Eq.{succ u3} β (MeasureTheory.SimpleFunc.toFun.{u1, u3} α _inst_1 β (MeasureTheory.SimpleFunc.approx.{u1, u3} α β _inst_1 _inst_2 _inst_3 _inst_4 i (Function.comp.{succ u1, succ u2, succ u3} α γ β f g) n) a) (MeasureTheory.SimpleFunc.toFun.{u2, u3} γ _inst_9 β (MeasureTheory.SimpleFunc.approx.{u2, u3} γ β _inst_9 _inst_2 _inst_3 _inst_4 i f n) (g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.approx_comp MeasureTheory.SimpleFunc.approx_compₓ'. -/ theorem approx_comp [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] [MeasurableSpace γ] {i : ℕ → β} {f : γ → β} {g : α → γ} {n : ℕ} (a : α) (hf : Measurable f) (hg : Measurable g) : @@ -832,6 +1324,12 @@ theorem approx_comp [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSp end +/- warning: measure_theory.simple_func.supr_approx_apply -> MeasureTheory.SimpleFunc.iSup_approx_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : CompleteLattice.{u2} β] [_inst_4 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_3)))] [_inst_5 : Zero.{u2} β] [_inst_6 : MeasurableSpace.{u2} β] [_inst_7 : OpensMeasurableSpace.{u2} β _inst_2 _inst_6] (i : Nat -> β) (f : α -> β) (a : α), (Measurable.{u1, u2} α β _inst_1 _inst_6 f) -> (Eq.{succ u2} β (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_5))) (Bot.bot.{u2} β (CompleteLattice.toHasBot.{u2} β _inst_3))) -> (Eq.{succ u2} β (iSup.{u2, 1} β (ConditionallyCompleteLattice.toHasSup.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) Nat (fun (n : Nat) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 (Lattice.toSemilatticeSup.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3))) (BoundedOrder.toOrderBot.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β (Lattice.toSemilatticeSup.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)))))) (CompleteLattice.toBoundedOrder.{u2} β _inst_3)) _inst_5 i f n) a)) (iSup.{u2, 1} β (ConditionallyCompleteLattice.toHasSup.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) Nat (fun (k : Nat) => iSup.{u2, 0} β (ConditionallyCompleteLattice.toHasSup.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_3)))) (i k) (f a)) (fun (h : LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (CompleteSemilatticeInf.toPartialOrder.{u2} β (CompleteLattice.toCompleteSemilatticeInf.{u2} β _inst_3)))) (i k) (f a)) => i k)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : CompleteLattice.{u2} β] [_inst_4 : OrderClosedTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (OmegaCompletePartialOrder.toPartialOrder.{u2} β (CompleteLattice.instOmegaCompletePartialOrder.{u2} β _inst_3)))] [_inst_5 : Zero.{u2} β] [_inst_6 : MeasurableSpace.{u2} β] [_inst_7 : OpensMeasurableSpace.{u2} β _inst_2 _inst_6] (i : Nat -> β) (f : α -> β) (a : α), (Measurable.{u1, u2} α β _inst_1 _inst_6 f) -> (Eq.{succ u2} β (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_5)) (Bot.bot.{u2} β (CompleteLattice.toBot.{u2} β _inst_3))) -> (Eq.{succ u2} β (iSup.{u2, 1} β (ConditionallyCompleteLattice.toSupSet.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) Nat (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, u2} α _inst_1 β (MeasureTheory.SimpleFunc.approx.{u1, u2} α β _inst_1 (Lattice.toSemilatticeSup.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3))) (BoundedOrder.toOrderBot.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β (Lattice.toSemilatticeSup.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)))))) (CompleteLattice.toBoundedOrder.{u2} β _inst_3)) _inst_5 i f n) a)) (iSup.{u2, 1} β (ConditionallyCompleteLattice.toSupSet.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) Nat (fun (k : Nat) => iSup.{u2, 0} β (ConditionallyCompleteLattice.toSupSet.{u2} β (CompleteLattice.toConditionallyCompleteLattice.{u2} β _inst_3)) (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (OmegaCompletePartialOrder.toPartialOrder.{u2} β (CompleteLattice.instOmegaCompletePartialOrder.{u2} β _inst_3)))) (i k) (f a)) (fun (h : LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (OmegaCompletePartialOrder.toPartialOrder.{u2} β (CompleteLattice.instOmegaCompletePartialOrder.{u2} β _inst_3)))) (i k) (f a)) => i k)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.supr_approx_apply MeasureTheory.SimpleFunc.iSup_approx_applyₓ'. -/ theorem iSup_approx_apply [TopologicalSpace β] [CompleteLattice β] [OrderClosedTopology β] [Zero β] [MeasurableSpace β] [OpensMeasurableSpace β] (i : ℕ → β) (f : α → β) (a : α) (hf : Measurable f) (h_zero : (0 : β) = ⊥) : (⨆ n, (approx i f n : α →ₛ β) a) = ⨆ (k) (h : i k ≤ f a), i k := @@ -853,20 +1351,36 @@ end Approx section Eapprox +#print MeasureTheory.SimpleFunc.ennrealRatEmbed /- /-- A sequence of `ℝ≥0∞`s such that its range is the set of non-negative rational numbers. -/ def ennrealRatEmbed (n : ℕ) : ℝ≥0∞ := ENNReal.ofReal ((Encodable.decode ℚ n).getD (0 : ℚ)) #align measure_theory.simple_func.ennreal_rat_embed MeasureTheory.SimpleFunc.ennrealRatEmbed +-/ +/- warning: measure_theory.simple_func.ennreal_rat_embed_encode -> MeasureTheory.SimpleFunc.ennrealRatEmbed_encode is a dubious translation: +lean 3 declaration is + forall (q : Rat), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.ennrealRatEmbed (Encodable.encode.{0} Rat Rat.encodable q)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (Real.toNNReal ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q))) +but is expected to have type + forall (q : Rat), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.ennrealRatEmbed (Encodable.encode.{0} Rat Rat.instEncodableRat q)) (ENNReal.some (Real.toNNReal (Rat.cast.{0} Real Real.ratCast q))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.ennreal_rat_embed_encode MeasureTheory.SimpleFunc.ennrealRatEmbed_encodeₓ'. -/ theorem ennrealRatEmbed_encode (q : ℚ) : ennrealRatEmbed (Encodable.encode q) = Real.toNNReal q := by rw [ennreal_rat_embed, Encodable.encodek] <;> rfl #align measure_theory.simple_func.ennreal_rat_embed_encode MeasureTheory.SimpleFunc.ennrealRatEmbed_encode +#print MeasureTheory.SimpleFunc.eapprox /- /-- Approximate a function `α → ℝ≥0∞` by a sequence of simple functions. -/ def eapprox : (α → ℝ≥0∞) → ℕ → α →ₛ ℝ≥0∞ := approx ennrealRatEmbed #align measure_theory.simple_func.eapprox MeasureTheory.SimpleFunc.eapprox +-/ +/- warning: measure_theory.simple_func.eapprox_lt_top -> MeasureTheory.SimpleFunc.eapprox_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal) (n : Nat) (a : α), LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal _inst_1) (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal) (n : Nat) (a : α), LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 ENNReal (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.eapprox_lt_top MeasureTheory.SimpleFunc.eapprox_lt_topₓ'. -/ theorem eapprox_lt_top (f : α → ℝ≥0∞) (n : ℕ) (a : α) : eapprox f n a < ∞ := by simp only [eapprox, approx, finset_sup_apply, Finset.sup_lt_iff, WithTop.zero_lt_top, @@ -883,11 +1397,23 @@ theorem eapprox_lt_top (f : α → ℝ≥0∞) (n : ℕ) (a : α) : eapprox f n · exact WithTop.zero_lt_top #align measure_theory.simple_func.eapprox_lt_top MeasureTheory.SimpleFunc.eapprox_lt_top +/- warning: measure_theory.simple_func.monotone_eapprox -> MeasureTheory.SimpleFunc.monotone_eapprox is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), Monotone.{0, u1} Nat (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (MeasureTheory.SimpleFunc.instPreorder.{u1, 0} α ENNReal _inst_1 (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), Monotone.{0, u1} Nat (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (MeasureTheory.SimpleFunc.instPreorder.{u1, 0} α ENNReal _inst_1 (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.monotone_eapprox MeasureTheory.SimpleFunc.monotone_eapproxₓ'. -/ @[mono] theorem monotone_eapprox (f : α → ℝ≥0∞) : Monotone (eapprox f) := monotone_approx _ f #align measure_theory.simple_func.monotone_eapprox MeasureTheory.SimpleFunc.monotone_eapprox +/- warning: measure_theory.simple_func.supr_eapprox_apply -> MeasureTheory.SimpleFunc.iSup_eapprox_apply is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal _inst_1) (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a)) (f a)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 ENNReal (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a)) (f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.supr_eapprox_apply MeasureTheory.SimpleFunc.iSup_eapprox_applyₓ'. -/ theorem iSup_eapprox_apply (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) : (⨆ n, (eapprox f n : α →ₛ ℝ≥0∞) a) = f a := by @@ -905,18 +1431,28 @@ theorem iSup_eapprox_apply (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) exact lt_irrefl _ (lt_of_le_of_lt this lt_q) #align measure_theory.simple_func.supr_eapprox_apply MeasureTheory.SimpleFunc.iSup_eapprox_apply +#print MeasureTheory.SimpleFunc.eapprox_comp /- theorem eapprox_comp [MeasurableSpace γ] {f : γ → ℝ≥0∞} {g : α → γ} {n : ℕ} (hf : Measurable f) (hg : Measurable g) : (eapprox (f ∘ g) n : α → ℝ≥0∞) = (eapprox f n : γ →ₛ ℝ≥0∞) ∘ g := funext fun a => approx_comp a hf hg #align measure_theory.simple_func.eapprox_comp MeasureTheory.SimpleFunc.eapprox_comp +-/ +#print MeasureTheory.SimpleFunc.eapproxDiff /- /-- Approximate a function `α → ℝ≥0∞` by a series of simple functions taking their values in `ℝ≥0`. -/ def eapproxDiff (f : α → ℝ≥0∞) : ∀ n : ℕ, α →ₛ ℝ≥0 | 0 => (eapprox f 0).map ENNReal.toNNReal | n + 1 => (eapprox f (n + 1) - eapprox f n).map ENNReal.toNNReal #align measure_theory.simple_func.eapprox_diff MeasureTheory.SimpleFunc.eapproxDiff +-/ +/- warning: measure_theory.simple_func.sum_eapprox_diff -> MeasureTheory.SimpleFunc.sum_eapproxDiff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal) (n : Nat) (a : α), Eq.{1} ENNReal (Finset.sum.{0, 0} ENNReal Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (k : Nat) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal _inst_1) (MeasureTheory.SimpleFunc.eapproxDiff.{u1} α _inst_1 f k) a))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal _inst_1) (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal) (n : Nat) (a : α), Eq.{1} ENNReal (Finset.sum.{0, 0} ENNReal Nat (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (k : Nat) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 NNReal (MeasureTheory.SimpleFunc.eapproxDiff.{u1} α _inst_1 f k) a))) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 ENNReal (MeasureTheory.SimpleFunc.eapprox.{u1} α _inst_1 f n) a) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.sum_eapprox_diff MeasureTheory.SimpleFunc.sum_eapproxDiffₓ'. -/ theorem sum_eapproxDiff (f : α → ℝ≥0∞) (n : ℕ) (a : α) : (∑ k in Finset.range (n + 1), (eapproxDiff f k a : ℝ≥0∞)) = eapprox f n a := by @@ -931,6 +1467,12 @@ theorem sum_eapproxDiff (f : α → ℝ≥0∞) (n : ℕ) (a : α) : exact le_self_add #align measure_theory.simple_func.sum_eapprox_diff MeasureTheory.SimpleFunc.sum_eapproxDiff +/- warning: measure_theory.simple_func.tsum_eapprox_diff -> MeasureTheory.SimpleFunc.tsum_eapproxDiff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (tsum.{0, 0} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace Nat (fun (n : Nat) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal _inst_1) (MeasureTheory.SimpleFunc.eapproxDiff.{u1} α _inst_1 f n) a))) (f a)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : α -> ENNReal), (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (tsum.{0, 0} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal Nat (fun (n : Nat) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 NNReal (MeasureTheory.SimpleFunc.eapproxDiff.{u1} α _inst_1 f n) a))) (f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.tsum_eapprox_diff MeasureTheory.SimpleFunc.tsum_eapproxDiffₓ'. -/ theorem tsum_eapproxDiff (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) : (∑' n, (eapproxDiff f n a : ℝ≥0∞)) = f a := by simp_rw [ENNReal.tsum_eq_iSup_nat' (tendsto_add_at_top_nat 1), sum_eapprox_diff, @@ -945,11 +1487,19 @@ section Measure variable {m : MeasurableSpace α} {μ ν : Measure α} +#print MeasureTheory.SimpleFunc.lintegral /- /-- Integral of a simple function whose codomain is `ℝ≥0∞`. -/ def lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ := ∑ x in f.range, x * μ (f ⁻¹' {x}) #align measure_theory.simple_func.lintegral MeasureTheory.SimpleFunc.lintegral +-/ +/- warning: measure_theory.simple_func.lintegral_eq_of_subset -> MeasureTheory.SimpleFunc.lintegral_eq_of_subset is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Finset.{0} ENNReal}, (forall (x : α), (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f x) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, 0} α ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.hasSingleton.{0} ENNReal) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f x)))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Membership.Mem.{0, 0} ENNReal (Finset.{0} ENNReal) (Finset.hasMem.{0} ENNReal) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f x) s)) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Finset.sum.{0, 0} ENNReal ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (x : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) x (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, 0} α ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.hasSingleton.{0} ENNReal) x)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Finset.{0} ENNReal}, (forall (x : α), (Ne.{1} ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.preimage.{u1, 0} α ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.instSingletonSet.{0} ENNReal) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f x)))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Membership.mem.{0, 0} ENNReal (Finset.{0} ENNReal) (Finset.instMembershipFinset.{0} ENNReal) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f x) s)) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Finset.sum.{0, 0} ENNReal ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (x : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) x (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.preimage.{u1, 0} α ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.instSingletonSet.{0} ENNReal) x)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_eq_of_subset MeasureTheory.SimpleFunc.lintegral_eq_of_subsetₓ'. -/ theorem lintegral_eq_of_subset (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : ∀ x, f x ≠ 0 → μ (f ⁻¹' {f x}) ≠ 0 → f x ∈ s) : f.lintegral μ = ∑ x in s, x * μ (f ⁻¹' {x}) := @@ -966,12 +1516,24 @@ theorem lintegral_eq_of_subset (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞ rfl #align measure_theory.simple_func.lintegral_eq_of_subset MeasureTheory.SimpleFunc.lintegral_eq_of_subset +/- warning: measure_theory.simple_func.lintegral_eq_of_subset' -> MeasureTheory.SimpleFunc.lintegral_eq_of_subset' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Finset.{0} ENNReal}, (HasSubset.Subset.{0} (Finset.{0} ENNReal) (Finset.hasSubset.{0} ENNReal) (SDiff.sdiff.{0} (Finset.{0} ENNReal) (Finset.hasSdiff.{0} ENNReal (fun (a : ENNReal) (b : ENNReal) => Option.decidableEq.{0} NNReal (fun (a : NNReal) (b : NNReal) => Subtype.decidableEq.{0} Real (fun (x : Real) => LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) x) (fun (a : Real) (b : Real) => Real.decidableEq a b) a b) a b)) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (Singleton.singleton.{0, 0} ENNReal (Finset.{0} ENNReal) (Finset.hasSingleton.{0} ENNReal) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))))) s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Finset.sum.{0, 0} ENNReal ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (x : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) x (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, 0} α ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.hasSingleton.{0} ENNReal) x)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Finset.{0} ENNReal}, (HasSubset.Subset.{0} (Finset.{0} ENNReal) (Finset.instHasSubsetFinset.{0} ENNReal) (SDiff.sdiff.{0} (Finset.{0} ENNReal) (Finset.instSDiffFinset.{0} ENNReal (fun (a : ENNReal) (b : ENNReal) => instDecidableEq.{0} ENNReal (instLinearOrder.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) a b)) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (Singleton.singleton.{0, 0} ENNReal (Finset.{0} ENNReal) (Finset.instSingletonFinset.{0} ENNReal) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)))) s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Finset.sum.{0, 0} ENNReal ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (x : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) x (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.preimage.{u1, 0} α ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.instSingletonSet.{0} ENNReal) x)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_eq_of_subset' MeasureTheory.SimpleFunc.lintegral_eq_of_subset'ₓ'. -/ theorem lintegral_eq_of_subset' (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : f.range \ {0} ⊆ s) : f.lintegral μ = ∑ x in s, x * μ (f ⁻¹' {x}) := f.lintegral_eq_of_subset fun x hfx _ => hs <| Finset.mem_sdiff.2 ⟨f.mem_range_self x, mt Finset.mem_singleton.1 hfx⟩ #align measure_theory.simple_func.lintegral_eq_of_subset' MeasureTheory.SimpleFunc.lintegral_eq_of_subset' +/- warning: measure_theory.simple_func.map_lintegral -> MeasureTheory.SimpleFunc.map_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (g : β -> ENNReal) (f : MeasureTheory.SimpleFunc.{u1, u2} α m β), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.map.{u1, u2, 0} α β ENNReal m g f) μ) (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.SimpleFunc.range.{u1, u2} α β m f) (fun (x : β) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (g x) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) x))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} (g : β -> ENNReal) (f : MeasureTheory.SimpleFunc.{u2, u1} α m β), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u2} α m (MeasureTheory.SimpleFunc.map.{u2, u1, 0} α β ENNReal m g f) μ) (Finset.sum.{0, u1} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.SimpleFunc.range.{u2, u1} α β m f) (fun (x : β) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (g x) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) x))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.map_lintegral MeasureTheory.SimpleFunc.map_lintegralₓ'. -/ /-- Calculate the integral of `(g ∘ f)`, where `g : β → ℝ≥0∞` and `f : α →ₛ β`. -/ theorem map_lintegral (g : β → ℝ≥0∞) (f : α →ₛ β) : (f.map g).lintegral μ = ∑ x in f.range, g x * μ (f ⁻¹' {x}) := @@ -988,6 +1550,12 @@ theorem map_lintegral (g : β → ℝ≥0∞) (f : α →ₛ β) : rw [h] #align measure_theory.simple_func.map_lintegral MeasureTheory.SimpleFunc.map_lintegral +/- warning: measure_theory.simple_func.add_lintegral -> MeasureTheory.SimpleFunc.add_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (instHAdd.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instAdd.{u1, 0} α ENNReal m (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g) μ) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (instHAdd.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instAdd.{u1, 0} α ENNReal m (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))))))) f g) μ) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.add_lintegral MeasureTheory.SimpleFunc.add_lintegralₓ'. -/ theorem add_lintegral (f g : α →ₛ ℝ≥0∞) : (f + g).lintegral μ = f.lintegral μ + g.lintegral μ := calc (f + g).lintegral μ = @@ -1003,6 +1571,12 @@ theorem add_lintegral (f g : α →ₛ ℝ≥0∞) : (f + g).lintegral μ = f.li #align measure_theory.simple_func.add_lintegral MeasureTheory.SimpleFunc.add_lintegral +/- warning: measure_theory.simple_func.const_mul_lintegral -> MeasureTheory.SimpleFunc.const_mul_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (x : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (HMul.hMul.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (instHMul.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instMul.{u1, 0} α ENNReal m (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m x) f) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) x (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (x : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (HMul.hMul.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (instHMul.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instMul.{u1, 0} α ENNReal m (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m x) f) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) x (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.const_mul_lintegral MeasureTheory.SimpleFunc.const_mul_lintegralₓ'. -/ theorem const_mul_lintegral (f : α →ₛ ℝ≥0∞) (x : ℝ≥0∞) : (const α x * f).lintegral μ = x * f.lintegral μ := calc @@ -1013,6 +1587,12 @@ theorem const_mul_lintegral (f : α →ₛ ℝ≥0∞) (x : ℝ≥0∞) : #align measure_theory.simple_func.const_mul_lintegral MeasureTheory.SimpleFunc.const_mul_lintegral +/- warning: measure_theory.simple_func.lintegralₗ -> MeasureTheory.SimpleFunc.lintegralₗ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α}, LinearMap.{0, 0, u1, u1} ENNReal ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (LinearMap.{0, 0, u1, 0} ENNReal ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (MeasureTheory.Measure.{u1} α m) ENNReal (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) MeasureTheory.SimpleFunc.lintegralₗ._proof_1 m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (MeasureTheory.SimpleFunc.instAddCommMonoid.{u1, 0} α ENNReal m (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (LinearMap.addCommMonoid.{0, 0, u1, 0} ENNReal ENNReal (MeasureTheory.Measure.{u1} α m) ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) MeasureTheory.SimpleFunc.lintegralₗ._proof_1 m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (MeasureTheory.SimpleFunc.instModule.{u1, 0, 0} α ENNReal m ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (LinearMap.module.{0, 0, 0, u1, 0} ENNReal ENNReal ENNReal (MeasureTheory.Measure.{u1} α m) ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) MeasureTheory.SimpleFunc.lintegralₗ._proof_1 m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) MeasureTheory.SimpleFunc.lintegralₗ._proof_2) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α}, LinearMap.{0, 0, u1, u1} ENNReal ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (LinearMap.{0, 0, u1, 0} ENNReal ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) (MeasureTheory.Measure.{u1} α m) ENNReal (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) (MeasureTheory.SimpleFunc.instAddCommMonoid.{u1, 0} α ENNReal m (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal))) (LinearMap.addCommMonoid.{0, 0, u1, 0} ENNReal ENNReal (MeasureTheory.Measure.{u1} α m) ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))) (MeasureTheory.SimpleFunc.instModule.{u1, 0, 0} α ENNReal m ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) (LinearMap.instModuleLinearMapAddCommMonoid.{0, 0, 0, u1, 0} ENNReal ENNReal ENNReal (MeasureTheory.Measure.{u1} α m) ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.Measure.instModule.{u1, 0} α ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (RingHom.id.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Semiring.toModule.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) (smulCommClass_self.{0, 0} ENNReal ENNReal (LinearOrderedCommMonoid.toCommMonoid.{0} ENNReal (LinearOrderedCommMonoidWithZero.toLinearOrderedCommMonoid.{0} ENNReal ENNReal.instLinearOrderedCommMonoidWithZeroENNReal)) (MulActionWithZero.toMulAction.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) instENNRealZero (MonoidWithZero.toMulActionWithZero.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegralₗ MeasureTheory.SimpleFunc.lintegralₗₓ'. -/ /-- Integral of a simple function `α →ₛ ℝ≥0∞` as a bilinear map. -/ def lintegralₗ {m : MeasurableSpace α} : (α →ₛ ℝ≥0∞) →ₗ[ℝ≥0∞] Measure α →ₗ[ℝ≥0∞] ℝ≥0∞ where @@ -1024,24 +1604,54 @@ def lintegralₗ {m : MeasurableSpace α} : (α →ₛ ℝ≥0∞) →ₗ[ℝ≥ map_smul' c f := LinearMap.ext fun μ => const_mul_lintegral f c #align measure_theory.simple_func.lintegralₗ MeasureTheory.SimpleFunc.lintegralₗ +/- warning: measure_theory.simple_func.zero_lintegral -> MeasureTheory.SimpleFunc.zero_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (OfNat.ofNat.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) 0 (OfNat.mk.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) 0 (Zero.zero.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instZero.{u1, 0} α ENNReal m ENNReal.hasZero)))) μ) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (OfNat.ofNat.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) 0 (Zero.toOfNat0.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instZero.{u1, 0} α ENNReal m instENNRealZero))) μ) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.zero_lintegral MeasureTheory.SimpleFunc.zero_lintegralₓ'. -/ @[simp] theorem zero_lintegral : (0 : α →ₛ ℝ≥0∞).lintegral μ = 0 := LinearMap.ext_iff.1 lintegralₗ.map_zero μ #align measure_theory.simple_func.zero_lintegral MeasureTheory.SimpleFunc.zero_lintegral +/- warning: measure_theory.simple_func.lintegral_add -> MeasureTheory.SimpleFunc.lintegral_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ν : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHAdd.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instAdd.{u1} α m)) μ ν)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f ν)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ν : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHAdd.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instAdd.{u1} α m)) μ ν)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f ν)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_add MeasureTheory.SimpleFunc.lintegral_addₓ'. -/ theorem lintegral_add {ν} (f : α →ₛ ℝ≥0∞) : f.lintegral (μ + ν) = f.lintegral μ + f.lintegral ν := (lintegralₗ f).map_add μ ν #align measure_theory.simple_func.lintegral_add MeasureTheory.SimpleFunc.lintegral_add +/- warning: measure_theory.simple_func.lintegral_smul -> MeasureTheory.SimpleFunc.lintegral_smul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) m) c μ)) (SMul.smul.{0, 0} ENNReal ENNReal (Mul.toSMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m)) c μ)) (HSMul.hSMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHSMul.{0, 0} ENNReal ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))) c (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_smul MeasureTheory.SimpleFunc.lintegral_smulₓ'. -/ theorem lintegral_smul (f : α →ₛ ℝ≥0∞) (c : ℝ≥0∞) : f.lintegral (c • μ) = c • f.lintegral μ := (lintegralₗ f).map_smul c μ #align measure_theory.simple_func.lintegral_smul MeasureTheory.SimpleFunc.lintegral_smul +/- warning: measure_theory.simple_func.lintegral_zero -> MeasureTheory.SimpleFunc.lintegral_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α _inst_1 f (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α _inst_1 f (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1)))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_zero MeasureTheory.SimpleFunc.lintegral_zeroₓ'. -/ @[simp] theorem lintegral_zero [MeasurableSpace α] (f : α →ₛ ℝ≥0∞) : f.lintegral 0 = 0 := (lintegralₗ f).map_zero #align measure_theory.simple_func.lintegral_zero MeasureTheory.SimpleFunc.lintegral_zero +/- warning: measure_theory.simple_func.lintegral_sum -> MeasureTheory.SimpleFunc.lintegral_sum is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {ι : Type.{u2}} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (MeasureTheory.Measure.sum.{u1, u2} α ι m μ)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace ι (fun (i : ι) => MeasureTheory.SimpleFunc.lintegral.{u1} α m f (μ i))) +but is expected to have type + forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {ι : Type.{u1}} (f : MeasureTheory.SimpleFunc.{u2, 0} α m ENNReal) (μ : ι -> (MeasureTheory.Measure.{u2} α m)), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u2} α m f (MeasureTheory.Measure.sum.{u2, u1} α ι m μ)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal ι (fun (i : ι) => MeasureTheory.SimpleFunc.lintegral.{u2} α m f (μ i))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_sum MeasureTheory.SimpleFunc.lintegral_sumₓ'. -/ theorem lintegral_sum {m : MeasurableSpace α} {ι} (f : α →ₛ ℝ≥0∞) (μ : ι → Measure α) : f.lintegral (Measure.sum μ) = ∑' i, f.lintegral (μ i) := by @@ -1050,6 +1660,12 @@ theorem lintegral_sum {m : MeasurableSpace α} {ι} (f : α →ₛ ℝ≥0∞) ( apply ENNReal.tsum_comm #align measure_theory.simple_func.lintegral_sum MeasureTheory.SimpleFunc.lintegral_sum +/- warning: measure_theory.simple_func.restrict_lintegral -> MeasureTheory.SimpleFunc.restrict_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m ENNReal.hasZero f s) μ) (Finset.sum.{0, 0} ENNReal ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (fun (r : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Set.preimage.{u1, 0} α ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.hasSingleton.{0} ENNReal) r)) s))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m instENNRealZero f s) μ) (Finset.sum.{0, 0} ENNReal ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (fun (r : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Set.preimage.{u1, 0} α ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.instSingletonSet.{0} ENNReal) r)) s))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_lintegral MeasureTheory.SimpleFunc.restrict_lintegralₓ'. -/ theorem restrict_lintegral (f : α →ₛ ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : (restrict f s).lintegral μ = ∑ r in f.range, r * μ (f ⁻¹' {r} ∩ s) := calc @@ -1066,16 +1682,34 @@ theorem restrict_lintegral (f : α →ₛ ℝ≥0∞) {s : Set α} (hs : Measura #align measure_theory.simple_func.restrict_lintegral MeasureTheory.SimpleFunc.restrict_lintegral +/- warning: measure_theory.simple_func.lintegral_restrict -> MeasureTheory.SimpleFunc.lintegral_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (s : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (MeasureTheory.Measure.restrict.{u1} α m μ s)) (Finset.sum.{0, 0} ENNReal ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (fun (y : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) y (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (Set.preimage.{u1, 0} α ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.hasSingleton.{0} ENNReal) y)) s)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (s : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (MeasureTheory.Measure.restrict.{u1} α m μ s)) (Finset.sum.{0, 0} ENNReal ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (MeasureTheory.SimpleFunc.range.{u1, 0} α ENNReal m f) (fun (y : ENNReal) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) y (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (Set.preimage.{u1, 0} α ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f) (Singleton.singleton.{0, 0} ENNReal (Set.{0} ENNReal) (Set.instSingletonSet.{0} ENNReal) y)) s)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_restrict MeasureTheory.SimpleFunc.lintegral_restrictₓ'. -/ theorem lintegral_restrict {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (s : Set α) (μ : Measure α) : f.lintegral (μ.restrict s) = ∑ y in f.range, y * μ (f ⁻¹' {y} ∩ s) := by simp only [lintegral, measure.restrict_apply, f.measurable_set_preimage] #align measure_theory.simple_func.lintegral_restrict MeasureTheory.SimpleFunc.lintegral_restrict +/- warning: measure_theory.simple_func.restrict_lintegral_eq_lintegral_restrict -> MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m ENNReal.hasZero f s) μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (MeasureTheory.Measure.restrict.{u1} α m μ s))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m instENNRealZero f s) μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f (MeasureTheory.Measure.restrict.{u1} α m μ s))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_lintegral_eq_lintegral_restrict MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrictₓ'. -/ theorem restrict_lintegral_eq_lintegral_restrict (f : α →ₛ ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : (restrict f s).lintegral μ = f.lintegral (μ.restrict s) := by rw [f.restrict_lintegral hs, lintegral_restrict] #align measure_theory.simple_func.restrict_lintegral_eq_lintegral_restrict MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict +/- warning: measure_theory.simple_func.const_lintegral -> MeasureTheory.SimpleFunc.const_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.const_lintegral MeasureTheory.SimpleFunc.const_lintegralₓ'. -/ theorem const_lintegral (c : ℝ≥0∞) : (const α c).lintegral μ = c * μ univ := by rw [lintegral] @@ -1084,16 +1718,34 @@ theorem const_lintegral (c : ℝ≥0∞) : (const α c).lintegral μ = c * μ un · simp [preimage_const_of_mem] #align measure_theory.simple_func.const_lintegral MeasureTheory.SimpleFunc.const_lintegral +/- warning: measure_theory.simple_func.const_lintegral_restrict -> MeasureTheory.SimpleFunc.const_lintegral_restrict is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) (s : Set.{u1} α), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) (MeasureTheory.Measure.restrict.{u1} α m μ s)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) (s : Set.{u1} α), Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) (MeasureTheory.Measure.restrict.{u1} α m μ s)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.const_lintegral_restrict MeasureTheory.SimpleFunc.const_lintegral_restrictₓ'. -/ theorem const_lintegral_restrict (c : ℝ≥0∞) (s : Set α) : (const α c).lintegral (μ.restrict s) = c * μ s := by rw [const_lintegral, measure.restrict_apply MeasurableSet.univ, univ_inter] #align measure_theory.simple_func.const_lintegral_restrict MeasureTheory.SimpleFunc.const_lintegral_restrict +/- warning: measure_theory.simple_func.restrict_const_lintegral -> MeasureTheory.SimpleFunc.restrict_const_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m ENNReal.hasZero (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) s) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.restrict.{u1, 0} α ENNReal m instENNRealZero (MeasureTheory.SimpleFunc.const.{u1, 0} α ENNReal m c) s) μ) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.restrict_const_lintegral MeasureTheory.SimpleFunc.restrict_const_lintegralₓ'. -/ theorem restrict_const_lintegral (c : ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : ((const α c).restrict s).lintegral μ = c * μ s := by rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict] #align measure_theory.simple_func.restrict_const_lintegral MeasureTheory.SimpleFunc.restrict_const_lintegral +/- warning: measure_theory.simple_func.le_sup_lintegral -> MeasureTheory.SimpleFunc.le_sup_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (Sup.sup.{0} ENNReal (SemilatticeSup.toHasSup.{0} ENNReal ENNReal.semilatticeSup) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g μ)) (MeasureTheory.SimpleFunc.lintegral.{u1} α m (Sup.sup.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instSup.{u1, 0} α ENNReal m (SemilatticeSup.toHasSup.{0} ENNReal ENNReal.semilatticeSup)) f g) μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (Sup.sup.{0} ENNReal (SemilatticeSup.toSup.{0} ENNReal instENNRealSemilatticeSup) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g μ)) (MeasureTheory.SimpleFunc.lintegral.{u1} α m (Sup.sup.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instSup.{u1, 0} α ENNReal m (SemilatticeSup.toSup.{0} ENNReal instENNRealSemilatticeSup)) f g) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.le_sup_lintegral MeasureTheory.SimpleFunc.le_sup_lintegralₓ'. -/ theorem le_sup_lintegral (f g : α →ₛ ℝ≥0∞) : f.lintegral μ ⊔ g.lintegral μ ≤ (f ⊔ g).lintegral μ := calc f.lintegral μ ⊔ g.lintegral μ = @@ -1109,6 +1761,12 @@ theorem le_sup_lintegral (f g : α →ₛ ℝ≥0∞) : f.lintegral μ ⊔ g.lin #align measure_theory.simple_func.le_sup_lintegral MeasureTheory.SimpleFunc.le_sup_lintegral +/- warning: measure_theory.simple_func.lintegral_mono -> MeasureTheory.SimpleFunc.lintegral_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ν : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal} {g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (LE.le.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instLE.{u1, 0} α ENNReal m (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) f g) -> (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toHasLe.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) μ ν) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g ν)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ν : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal} {g : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (LE.le.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (MeasureTheory.SimpleFunc.instLE.{u1, 0} α ENNReal m (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) f g) -> (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toLE.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) μ ν) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (MeasureTheory.SimpleFunc.lintegral.{u1} α m g ν)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.lintegral_mono MeasureTheory.SimpleFunc.lintegral_monoₓ'. -/ /-- `simple_func.lintegral` is monotone both in function and in measure. -/ @[mono] theorem lintegral_mono {f g : α →ₛ ℝ≥0∞} (hfg : f ≤ g) (hμν : μ ≤ ν) : @@ -1122,6 +1780,7 @@ theorem lintegral_mono {f g : α →ₛ ℝ≥0∞} (hfg : f ≤ g) (hμν : μ #align measure_theory.simple_func.lintegral_mono MeasureTheory.SimpleFunc.lintegral_mono +#print MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage /- /-- `simple_func.lintegral` depends only on the measures of `f ⁻¹' {y}`. -/ theorem lintegral_eq_of_measure_preimage [MeasurableSpace β] {f : α →ₛ ℝ≥0∞} {g : β →ₛ ℝ≥0∞} {ν : Measure β} (H : ∀ y, μ (f ⁻¹' {y}) = ν (g ⁻¹' {y})) : f.lintegral μ = g.lintegral ν := @@ -1132,13 +1791,17 @@ theorem lintegral_eq_of_measure_preimage [MeasurableSpace β] {f : α →ₛ ℝ intros exact mem_range_of_measure_ne_zero ‹_› #align measure_theory.simple_func.lintegral_eq_of_measure_preimage MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage +-/ +#print MeasureTheory.SimpleFunc.lintegral_congr /- /-- If two simple functions are equal a.e., then their `lintegral`s are equal. -/ theorem lintegral_congr {f g : α →ₛ ℝ≥0∞} (h : f =ᵐ[μ] g) : f.lintegral μ = g.lintegral μ := lintegral_eq_of_measure_preimage fun y => measure_congr <| Eventually.set_eq <| h.mono fun x hx => by simp [hx] #align measure_theory.simple_func.lintegral_congr MeasureTheory.SimpleFunc.lintegral_congr +-/ +#print MeasureTheory.SimpleFunc.lintegral_map' /- theorem lintegral_map' {β} [MeasurableSpace β] {μ' : Measure β} (f : α →ₛ ℝ≥0∞) (g : β →ₛ ℝ≥0∞) (m' : α → β) (eq : ∀ a, f a = g (m' a)) (h : ∀ s, MeasurableSet s → μ' s = μ (m' ⁻¹' s)) : f.lintegral μ = g.lintegral μ' := @@ -1147,11 +1810,14 @@ theorem lintegral_map' {β} [MeasurableSpace β] {μ' : Measure β} (f : α → simp only [preimage, Eq] exact (h (g ⁻¹' {y}) (g.measurable_set_preimage _)).symm #align measure_theory.simple_func.lintegral_map' MeasureTheory.SimpleFunc.lintegral_map' +-/ +#print MeasureTheory.SimpleFunc.lintegral_map /- theorem lintegral_map {β} [MeasurableSpace β] (g : β →ₛ ℝ≥0∞) {f : α → β} (hf : Measurable f) : g.lintegral (Measure.map f μ) = (g.comp f hf).lintegral μ := Eq.symm <| lintegral_map' _ _ f (fun a => rfl) fun s hs => Measure.map_apply hf hs #align measure_theory.simple_func.lintegral_map MeasureTheory.SimpleFunc.lintegral_map +-/ end Measure @@ -1159,6 +1825,12 @@ section FinMeasSupp open Finset Function +/- warning: measure_theory.simple_func.support_eq -> MeasureTheory.SimpleFunc.support_eq is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : Zero.{u2} β] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β), Eq.{succ u1} (Set.{u1} α) (Function.support.{u1, u2} α β _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f)) (Set.iUnion.{u1, succ u2} α β (fun (y : β) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) y (Finset.filter.{u2} β (fun (y : β) => Ne.{succ u2} β y (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) (fun (a : β) => Ne.decidable.{succ u2} β (fun (a : β) (b : β) => Classical.propDecidable (Eq.{succ u2} β a b)) a (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f))) (fun (H : Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) y (Finset.filter.{u2} β (fun (y : β) => Ne.{succ u2} β y (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) (fun (a : β) => Ne.decidable.{succ u2} β (fun (a : β) (b : β) => Classical.propDecidable (Eq.{succ u2} β a b)) a (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_2)))) (MeasureTheory.SimpleFunc.range.{u1, u2} α β _inst_1 f))) => Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_1) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) y)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : Zero.{u1} β] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 β), Eq.{succ u2} (Set.{u2} α) (Function.support.{u2, u1} α β _inst_2 (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f)) (Set.iUnion.{u2, succ u1} α β (fun (y : β) => Set.iUnion.{u2, 0} α (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) y (Finset.filter.{u1} β (fun (y : β) => Ne.{succ u1} β y (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) (fun (a : β) => instDecidableNot (Eq.{succ u1} β a (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) (Classical.propDecidable (Eq.{succ u1} β a (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))))) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f))) (fun (H : Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) y (Finset.filter.{u1} β (fun (y : β) => Ne.{succ u1} β y (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) (fun (a : β) => instDecidableNot (Eq.{succ u1} β a (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))) (Classical.propDecidable (Eq.{succ u1} β a (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_2))))) (MeasureTheory.SimpleFunc.range.{u2, u1} α β _inst_1 f))) => Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) y)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.support_eq MeasureTheory.SimpleFunc.support_eqₓ'. -/ theorem support_eq [MeasurableSpace α] [Zero β] (f : α →ₛ β) : support f = ⋃ y ∈ f.range.filterₓ fun y => y ≠ 0, f ⁻¹' {y} := Set.ext fun x => by @@ -1168,22 +1840,42 @@ theorem support_eq [MeasurableSpace α] [Zero β] (f : α →ₛ β) : variable {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : Measure α} {f : α →ₛ β} +/- warning: measure_theory.simple_func.measurable_set_support -> MeasureTheory.SimpleFunc.measurableSet_support is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Zero.{u2} β] [_inst_3 : MeasurableSpace.{u1} α] (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_3 β), MeasurableSet.{u1} α _inst_3 (Function.support.{u1, u2} α β _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_3 β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_3 β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β _inst_3) f)) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Zero.{u1} β] [_inst_3 : MeasurableSpace.{u2} α] (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_3 β), MeasurableSet.{u2} α _inst_3 (Function.support.{u2, u1} α β _inst_1 (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_3 β f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.measurable_set_support MeasureTheory.SimpleFunc.measurableSet_supportₓ'. -/ theorem measurableSet_support [MeasurableSpace α] (f : α →ₛ β) : MeasurableSet (support f) := by rw [f.support_eq] exact Finset.measurableSet_biUnion _ fun y hy => measurable_set_fiber _ _ #align measure_theory.simple_func.measurable_set_support MeasureTheory.SimpleFunc.measurableSet_support +#print MeasureTheory.SimpleFunc.FinMeasSupp /- /-- A `simple_func` has finite measure support if it is equal to `0` outside of a set of finite measure. -/ protected def FinMeasSupp {m : MeasurableSpace α} (f : α →ₛ β) (μ : Measure α) : Prop := f =ᶠ[μ.cofinite] 0 #align measure_theory.simple_func.fin_meas_supp MeasureTheory.SimpleFunc.FinMeasSupp +-/ +/- warning: measure_theory.simple_func.fin_meas_supp_iff_support -> MeasureTheory.SimpleFunc.finMeasSupp_iff_support is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β}, Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Function.support.{u1, u2} α β _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_1 : Zero.{u1} β] {μ : MeasureTheory.Measure.{u2} α m} {f : MeasureTheory.SimpleFunc.{u2, u1} α m β}, Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u2, u1} α β _inst_1 m f μ) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m μ) (Function.support.{u2, u1} α β _inst_1 (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp_iff_support MeasureTheory.SimpleFunc.finMeasSupp_iff_supportₓ'. -/ theorem finMeasSupp_iff_support : f.FinMeasSupp μ ↔ μ (support f) < ∞ := Iff.rfl #align measure_theory.simple_func.fin_meas_supp_iff_support MeasureTheory.SimpleFunc.finMeasSupp_iff_support +/- warning: measure_theory.simple_func.fin_meas_supp_iff -> MeasureTheory.SimpleFunc.finMeasSupp_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β}, Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) (forall (y : β), (Ne.{succ u2} β y (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_1)))) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) y))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_1 : Zero.{u1} β] {μ : MeasureTheory.Measure.{u2} α m} {f : MeasureTheory.SimpleFunc.{u2, u1} α m β}, Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u2, u1} α β _inst_1 m f μ) (forall (y : β), (Ne.{succ u1} β y (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_1))) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) y))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp_iff MeasureTheory.SimpleFunc.finMeasSupp_iffₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y «expr ≠ » 0) -/ theorem finMeasSupp_iff : f.FinMeasSupp μ ↔ ∀ (y) (_ : y ≠ 0), μ (f ⁻¹' {y}) < ∞ := by @@ -1198,25 +1890,51 @@ theorem finMeasSupp_iff : f.FinMeasSupp μ ↔ ∀ (y) (_ : y ≠ 0), μ (f ⁻ namespace FinMeasSupp +/- warning: measure_theory.simple_func.fin_meas_supp.meas_preimage_singleton_ne_zero -> MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) -> (forall {y : β}, (Ne.{succ u2} β y (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_1)))) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, u2} α β (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α m β) => α -> β) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α β m) f) (Singleton.singleton.{u2, u2} β (Set.{u2} β) (Set.hasSingleton.{u2} β) y))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} [_inst_1 : Zero.{u1} β] {μ : MeasureTheory.Measure.{u2} α m} {f : MeasureTheory.SimpleFunc.{u2, u1} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u2, u1} α β _inst_1 m f μ) -> (forall {y : β}, (Ne.{succ u1} β y (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_1))) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u2} α (MeasureTheory.Measure.toOuterMeasure.{u2} α m μ) (Set.preimage.{u2, u1} α β (MeasureTheory.SimpleFunc.toFun.{u2, u1} α m β f) (Singleton.singleton.{u1, u1} β (Set.{u1} β) (Set.instSingletonSet.{u1} β) y))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.meas_preimage_singleton_ne_zero MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zeroₓ'. -/ theorem meas_preimage_singleton_ne_zero (h : f.FinMeasSupp μ) {y : β} (hy : y ≠ 0) : μ (f ⁻¹' {y}) < ∞ := finMeasSupp_iff.1 h y hy #align measure_theory.simple_func.fin_meas_supp.meas_preimage_singleton_ne_zero MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero +/- warning: measure_theory.simple_func.fin_meas_supp.map -> MeasureTheory.SimpleFunc.FinMeasSupp.map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u3} γ] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : β -> γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) -> (Eq.{succ u3} γ (g (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_1)))) (OfNat.ofNat.{u3} γ 0 (OfNat.mk.{u3} γ 0 (Zero.zero.{u3} γ _inst_2)))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u3} α γ _inst_2 m (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ m g f) μ) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u1} γ] {μ : MeasureTheory.Measure.{u3} α m} {f : MeasureTheory.SimpleFunc.{u3, u2} α m β} {g : β -> γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u2} α β _inst_1 m f μ) -> (Eq.{succ u1} γ (g (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_1))) (OfNat.ofNat.{u1} γ 0 (Zero.toOfNat0.{u1} γ _inst_2))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u1} α γ _inst_2 m (MeasureTheory.SimpleFunc.map.{u3, u2, u1} α β γ m g f) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.map MeasureTheory.SimpleFunc.FinMeasSupp.mapₓ'. -/ protected theorem map {g : β → γ} (hf : f.FinMeasSupp μ) (hg : g 0 = 0) : (f.map g).FinMeasSupp μ := flip lt_of_le_of_lt hf (measure_mono <| support_comp_subset hg f) #align measure_theory.simple_func.fin_meas_supp.map MeasureTheory.SimpleFunc.FinMeasSupp.map +/- warning: measure_theory.simple_func.fin_meas_supp.of_map -> MeasureTheory.SimpleFunc.FinMeasSupp.of_map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u3} γ] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : β -> γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u3} α γ _inst_2 m (MeasureTheory.SimpleFunc.map.{u1, u2, u3} α β γ m g f) μ) -> (forall (b : β), (Eq.{succ u3} γ (g b) (OfNat.ofNat.{u3} γ 0 (OfNat.mk.{u3} γ 0 (Zero.zero.{u3} γ _inst_2)))) -> (Eq.{succ u2} β b (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_1))))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u1}} {γ : Type.{u2}} {m : MeasurableSpace.{u3} α} [_inst_1 : Zero.{u1} β] [_inst_2 : Zero.{u2} γ] {μ : MeasureTheory.Measure.{u3} α m} {f : MeasureTheory.SimpleFunc.{u3, u1} α m β} {g : β -> γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u2} α γ _inst_2 m (MeasureTheory.SimpleFunc.map.{u3, u1, u2} α β γ m g f) μ) -> (forall (b : β), (Eq.{succ u2} γ (g b) (OfNat.ofNat.{u2} γ 0 (Zero.toOfNat0.{u2} γ _inst_2))) -> (Eq.{succ u1} β b (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β _inst_1)))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u1} α β _inst_1 m f μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.of_map MeasureTheory.SimpleFunc.FinMeasSupp.of_mapₓ'. -/ theorem of_map {g : β → γ} (h : (f.map g).FinMeasSupp μ) (hg : ∀ b, g b = 0 → b = 0) : f.FinMeasSupp μ := flip lt_of_le_of_lt h <| measure_mono <| support_subset_comp hg _ #align measure_theory.simple_func.fin_meas_supp.of_map MeasureTheory.SimpleFunc.FinMeasSupp.of_map +#print MeasureTheory.SimpleFunc.FinMeasSupp.map_iff /- theorem map_iff {g : β → γ} (hg : ∀ {b}, g b = 0 ↔ b = 0) : (f.map g).FinMeasSupp μ ↔ f.FinMeasSupp μ := ⟨fun h => h.of_map fun b => hg.1, fun h => h.map <| hg.2 rfl⟩ #align measure_theory.simple_func.fin_meas_supp.map_iff MeasureTheory.SimpleFunc.FinMeasSupp.map_iff +-/ +/- warning: measure_theory.simple_func.fin_meas_supp.pair -> MeasureTheory.SimpleFunc.FinMeasSupp.pair is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u3} γ] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : MeasureTheory.SimpleFunc.{u1, u3} α m γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u3} α γ _inst_2 m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, max u2 u3} α (Prod.{u2, u3} β γ) (Prod.hasZero.{u2, u3} β γ _inst_1 _inst_2) m (MeasureTheory.SimpleFunc.pair.{u1, u2, u3} α β γ m f g) μ) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u1}} {γ : Type.{u2}} {m : MeasurableSpace.{u3} α} [_inst_1 : Zero.{u1} β] [_inst_2 : Zero.{u2} γ] {μ : MeasureTheory.Measure.{u3} α m} {f : MeasureTheory.SimpleFunc.{u3, u1} α m β} {g : MeasureTheory.SimpleFunc.{u3, u2} α m γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u1} α β _inst_1 m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u2} α γ _inst_2 m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, max u1 u2} α (Prod.{u1, u2} β γ) (Prod.instZeroSum.{u1, u2} β γ _inst_1 _inst_2) m (MeasureTheory.SimpleFunc.pair.{u3, u1, u2} α β γ m f g) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.pair MeasureTheory.SimpleFunc.FinMeasSupp.pairₓ'. -/ protected theorem pair {g : α →ₛ γ} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ) : (pair f g).FinMeasSupp μ := calc @@ -1226,11 +1944,23 @@ protected theorem pair {g : α →ₛ γ} (hf : f.FinMeasSupp μ) (hg : g.FinMea #align measure_theory.simple_func.fin_meas_supp.pair MeasureTheory.SimpleFunc.FinMeasSupp.pair +/- warning: measure_theory.simple_func.fin_meas_supp.map₂ -> MeasureTheory.SimpleFunc.FinMeasSupp.map₂ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {δ : Type.{u4}} {m : MeasurableSpace.{u1} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u3} γ] {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} [_inst_3 : Zero.{u4} δ], (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β _inst_1 m f μ) -> (forall {g : MeasureTheory.SimpleFunc.{u1, u3} α m γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u3} α γ _inst_2 m g μ) -> (forall {op : β -> γ -> δ}, (Eq.{succ u4} δ (op (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β _inst_1))) (OfNat.ofNat.{u3} γ 0 (OfNat.mk.{u3} γ 0 (Zero.zero.{u3} γ _inst_2)))) (OfNat.ofNat.{u4} δ 0 (OfNat.mk.{u4} δ 0 (Zero.zero.{u4} δ _inst_3)))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u4} α δ _inst_3 m (MeasureTheory.SimpleFunc.map.{u1, max u2 u3, u4} α (Prod.{u2, u3} β γ) δ m (Function.uncurry.{u2, u3, u4} β γ δ op) (MeasureTheory.SimpleFunc.pair.{u1, u2, u3} α β γ m f g)) μ))) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {δ : Type.{u4}} {m : MeasurableSpace.{u3} α} [_inst_1 : Zero.{u2} β] [_inst_2 : Zero.{u1} γ] {μ : MeasureTheory.Measure.{u3} α m} {f : MeasureTheory.SimpleFunc.{u3, u2} α m β} [_inst_3 : Zero.{u4} δ], (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u2} α β _inst_1 m f μ) -> (forall {g : MeasureTheory.SimpleFunc.{u3, u1} α m γ}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u1} α γ _inst_2 m g μ) -> (forall {op : β -> γ -> δ}, (Eq.{succ u4} δ (op (OfNat.ofNat.{u2} β 0 (Zero.toOfNat0.{u2} β _inst_1)) (OfNat.ofNat.{u1} γ 0 (Zero.toOfNat0.{u1} γ _inst_2))) (OfNat.ofNat.{u4} δ 0 (Zero.toOfNat0.{u4} δ _inst_3))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u3, u4} α δ _inst_3 m (MeasureTheory.SimpleFunc.map.{u3, max u1 u2, u4} α (Prod.{u2, u1} β γ) δ m (Function.uncurry.{u2, u1, u4} β γ δ op) (MeasureTheory.SimpleFunc.pair.{u3, u2, u1} α β γ m f g)) μ))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.map₂ MeasureTheory.SimpleFunc.FinMeasSupp.map₂ₓ'. -/ protected theorem map₂ [Zero δ] (hf : f.FinMeasSupp μ) {g : α →ₛ γ} (hg : g.FinMeasSupp μ) {op : β → γ → δ} (H : op 0 0 = 0) : ((pair f g).map (Function.uncurry op)).FinMeasSupp μ := (hf.pair hg).map H #align measure_theory.simple_func.fin_meas_supp.map₂ MeasureTheory.SimpleFunc.FinMeasSupp.map₂ +/- warning: measure_theory.simple_func.fin_meas_supp.add -> MeasureTheory.SimpleFunc.FinMeasSupp.add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_3 : AddMonoid.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : MeasureTheory.SimpleFunc.{u1, u2} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (instHAdd.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.instAdd.{u1, u2} α β m (AddZeroClass.toHasAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_3 : AddMonoid.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : MeasureTheory.SimpleFunc.{u1, u2} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddMonoid.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddMonoid.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (AddMonoid.toZero.{u2} β _inst_3) m (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (instHAdd.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.instAdd.{u1, u2} α β m (AddZeroClass.toAdd.{u2} β (AddMonoid.toAddZeroClass.{u2} β _inst_3)))) f g) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.add MeasureTheory.SimpleFunc.FinMeasSupp.addₓ'. -/ protected theorem add {β} [AddMonoid β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ) : (f + g).FinMeasSupp μ := by @@ -1238,6 +1968,12 @@ protected theorem add {β} [AddMonoid β] {f g : α →ₛ β} (hf : f.FinMeasSu exact hf.map₂ hg (zero_add 0) #align measure_theory.simple_func.fin_meas_supp.add MeasureTheory.SimpleFunc.FinMeasSupp.add +/- warning: measure_theory.simple_func.fin_meas_supp.mul -> MeasureTheory.SimpleFunc.FinMeasSupp.mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_3 : MonoidWithZero.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : MeasureTheory.SimpleFunc.{u1, u2} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MulZeroClass.toHasZero.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (instHMul.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.instMul.{u1, u2} α β m (MulZeroClass.toHasMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_3 : MonoidWithZero.{u2} β] {f : MeasureTheory.SimpleFunc.{u1, u2} α m β} {g : MeasureTheory.SimpleFunc.{u1, u2} α m β}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MonoidWithZero.toZero.{u2} β _inst_3) m f μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MonoidWithZero.toZero.{u2} β _inst_3) m g μ) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, u2} α β (MonoidWithZero.toZero.{u2} β _inst_3) m (HMul.hMul.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.{u1, u2} α m β) (instHMul.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α m β) (MeasureTheory.SimpleFunc.instMul.{u1, u2} α β m (MulZeroClass.toMul.{u2} β (MulZeroOneClass.toMulZeroClass.{u2} β (MonoidWithZero.toMulZeroOneClass.{u2} β _inst_3))))) f g) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.mul MeasureTheory.SimpleFunc.FinMeasSupp.mulₓ'. -/ protected theorem mul {β} [MonoidWithZero β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ) : (f * g).FinMeasSupp μ := by @@ -1245,6 +1981,12 @@ protected theorem mul {β} [MonoidWithZero β] {f g : α →ₛ β} (hf : f.FinM exact hf.map₂ hg (MulZeroClass.zero_mul 0) #align measure_theory.simple_func.fin_meas_supp.mul MeasureTheory.SimpleFunc.FinMeasSupp.mul +/- warning: measure_theory.simple_func.fin_meas_supp.lintegral_lt_top -> MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal ENNReal.hasZero m f μ) -> (Filter.Eventually.{u1} α (fun (a : α) => Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f a) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal instENNRealZero m f μ) -> (Filter.Eventually.{u1} α (fun (a : α) => Ne.{1} ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f a) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.lintegral_lt_top MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_topₓ'. -/ theorem lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf : ∀ᵐ a ∂μ, f a ≠ ∞) : f.lintegral μ < ∞ := by refine' sum_lt_top fun a ha => _ @@ -1257,6 +1999,12 @@ theorem lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf · exact mul_ne_top ha (fin_meas_supp_iff.1 hm _ ha0).Ne #align measure_theory.simple_func.fin_meas_supp.lintegral_lt_top MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top +/- warning: measure_theory.simple_func.fin_meas_supp.of_lintegral_ne_top -> MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (Ne.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal ENNReal.hasZero m f μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (Ne.{1} ENNReal (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal instENNRealZero m f μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.of_lintegral_ne_top MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_topₓ'. -/ theorem of_lintegral_ne_top {f : α →ₛ ℝ≥0∞} (h : f.lintegral μ ≠ ∞) : f.FinMeasSupp μ := by refine' fin_meas_supp_iff.2 fun b hb => _ @@ -1265,6 +2013,12 @@ theorem of_lintegral_ne_top {f : α →ₛ ℝ≥0∞} (h : f.lintegral μ ≠ exact (lt_top_of_sum_ne_top h (Finset.mem_insert_self _ _)).Ne #align measure_theory.simple_func.fin_meas_supp.of_lintegral_ne_top MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_top +/- warning: measure_theory.simple_func.fin_meas_supp.iff_lintegral_lt_top -> MeasureTheory.SimpleFunc.FinMeasSupp.iff_lintegral_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (Filter.Eventually.{u1} α (fun (a : α) => Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal) => α -> ENNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α ENNReal m) f a) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal ENNReal.hasZero m f μ) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : MeasureTheory.SimpleFunc.{u1, 0} α m ENNReal}, (Filter.Eventually.{u1} α (fun (a : α) => Ne.{1} ENNReal (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m ENNReal f a) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Iff (MeasureTheory.SimpleFunc.FinMeasSupp.{u1, 0} α ENNReal instENNRealZero m f μ) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m f μ) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.fin_meas_supp.iff_lintegral_lt_top MeasureTheory.SimpleFunc.FinMeasSupp.iff_lintegral_lt_topₓ'. -/ theorem iff_lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hf : ∀ᵐ a ∂μ, f a ≠ ∞) : f.FinMeasSupp μ ↔ f.lintegral μ < ∞ := ⟨fun h => h.lintegral_lt_top hf, fun h => of_lintegral_ne_top h.Ne⟩ @@ -1274,6 +2028,12 @@ end FinMeasSupp end FinMeasSupp +/- warning: measure_theory.simple_func.induction -> MeasureTheory.SimpleFunc.induction is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {γ : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : AddMonoid.{u2} γ] {P : (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) -> Prop}, (forall (c : γ) {s : Set.{u1} α} (hs : MeasurableSet.{u1} α _inst_1 s), P (MeasureTheory.SimpleFunc.piecewise.{u1, u2} α γ _inst_1 s hs (MeasureTheory.SimpleFunc.const.{u1, u2} α γ _inst_1 c) (MeasureTheory.SimpleFunc.const.{u1, u2} α γ _inst_1 (OfNat.ofNat.{u2} γ 0 (OfNat.mk.{u2} γ 0 (Zero.zero.{u2} γ (AddZeroClass.toHasZero.{u2} γ (AddMonoid.toAddZeroClass.{u2} γ _inst_2)))))))) -> (forall {{f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ}} {{g : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ}}, (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Function.support.{u1, u2} α γ (AddZeroClass.toHasZero.{u2} γ (AddMonoid.toAddZeroClass.{u2} γ _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α γ _inst_1) f)) (Function.support.{u1, u2} α γ (AddZeroClass.toHasZero.{u2} γ (AddMonoid.toAddZeroClass.{u2} γ _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (fun (_x : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) => α -> γ) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u2} α γ _inst_1) g))) -> (P f) -> (P g) -> (P (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (instHAdd.{max u1 u2} (MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ) (MeasureTheory.SimpleFunc.instAdd.{u1, u2} α γ _inst_1 (AddZeroClass.toHasAdd.{u2} γ (AddMonoid.toAddZeroClass.{u2} γ _inst_2)))) f g))) -> (forall (f : MeasureTheory.SimpleFunc.{u1, u2} α _inst_1 γ), P f) +but is expected to have type + forall {α : Type.{u2}} {γ : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : AddMonoid.{u1} γ] {P : (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) -> Prop}, (forall (c : γ) {s : Set.{u2} α} (hs : MeasurableSet.{u2} α _inst_1 s), P (MeasureTheory.SimpleFunc.piecewise.{u2, u1} α γ _inst_1 s hs (MeasureTheory.SimpleFunc.const.{u2, u1} α γ _inst_1 c) (MeasureTheory.SimpleFunc.const.{u2, u1} α γ _inst_1 (OfNat.ofNat.{u1} γ 0 (Zero.toOfNat0.{u1} γ (AddMonoid.toZero.{u1} γ _inst_2)))))) -> (forall {{f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ}} {{g : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ}}, (Disjoint.{u2} (Set.{u2} α) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} α) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} α) (Preorder.toLE.{u2} (Set.{u2} α) (PartialOrder.toPreorder.{u2} (Set.{u2} α) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} α) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))) (Function.support.{u2, u1} α γ (AddMonoid.toZero.{u1} γ _inst_2) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 γ f)) (Function.support.{u2, u1} α γ (AddMonoid.toZero.{u1} γ _inst_2) (MeasureTheory.SimpleFunc.toFun.{u2, u1} α _inst_1 γ g))) -> (P f) -> (P g) -> (P (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) (instHAdd.{max u2 u1} (MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ) (MeasureTheory.SimpleFunc.instAdd.{u2, u1} α γ _inst_1 (AddZeroClass.toAdd.{u1} γ (AddMonoid.toAddZeroClass.{u1} γ _inst_2)))) f g))) -> (forall (f : MeasureTheory.SimpleFunc.{u2, u1} α _inst_1 γ), P f) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.induction MeasureTheory.SimpleFunc.inductionₓ'. -/ /-- To prove something for an arbitrary simple function, it suffices to show that the property holds for (multiples of) characteristic functions and is closed under addition (of functions with disjoint support). @@ -1323,6 +2083,12 @@ end MeasureTheory open MeasureTheory MeasureTheory.SimpleFunc +/- warning: measurable.ennreal_induction -> Measurable.ennreal_induction is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {P : (α -> ENNReal) -> Prop}, (forall (c : ENNReal) {{s : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 s) -> (P (Set.indicator.{u1, 0} α ENNReal ENNReal.hasZero s (fun (_x : α) => c)))) -> (forall {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (Function.support.{u1, 0} α ENNReal ENNReal.hasZero f) (Function.support.{u1, 0} α ENNReal ENNReal.hasZero g)) -> (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace g) -> (P f) -> (P g) -> (P (HAdd.hAdd.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHAdd.{u1} (α -> ENNReal) (Pi.instAdd.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g))) -> (forall {{f : Nat -> α -> ENNReal}}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace (f n)) -> (Monotone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) f) -> (forall (n : Nat), P (f n)) -> (P (fun (x : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n x)))) -> (forall {{f : α -> ENNReal}}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (P f)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {P : (α -> ENNReal) -> Prop}, (forall (c : ENNReal) {{s : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 s) -> (P (Set.indicator.{u1, 0} α ENNReal instENNRealZero s (fun (_x : α) => c)))) -> (forall {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (Disjoint.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Function.support.{u1, 0} α ENNReal instENNRealZero f) (Function.support.{u1, 0} α ENNReal instENNRealZero g)) -> (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace g) -> (P f) -> (P g) -> (P (HAdd.hAdd.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHAdd.{u1} (α -> ENNReal) (Pi.instAdd.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))))))) f g))) -> (forall {{f : Nat -> α -> ENNReal}}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace (f n)) -> (Monotone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) f) -> (forall (n : Nat), P (f n)) -> (P (fun (x : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n x)))) -> (forall {{f : α -> ENNReal}}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (P f)) +Case conversion may be inaccurate. Consider using '#align measurable.ennreal_induction Measurable.ennreal_inductionₓ'. -/ /-- To prove something for an arbitrary measurable function into `ℝ≥0∞`, it suffices to show that the property holds for (multiples of) characteristic functions and is closed under addition and supremum of increasing sequences of functions. @@ -1332,7 +2098,7 @@ can be added once we need them (for example in `h_add` it is only necessary to c a simple function with a multiple of a characteristic function and that the intersection of their images is a subset of `{0}`. -/ @[elab_as_elim] -theorem Measurable.eNNReal_induction {α} [MeasurableSpace α] {P : (α → ℝ≥0∞) → Prop} +theorem Measurable.ennreal_induction {α} [MeasurableSpace α] {P : (α → ℝ≥0∞) → Prop} (h_ind : ∀ (c : ℝ≥0∞) ⦃s⦄, MeasurableSet s → P (indicator s fun _ => c)) (h_add : ∀ ⦃f g : α → ℝ≥0∞⦄, @@ -1349,5 +2115,5 @@ theorem Measurable.eNNReal_induction {α} [MeasurableSpace α] {P : (α → ℝ exact fun n => simple_func.induction (fun c s hs => h_ind c hs) (fun f g hfg hf hg => h_add hfg f.Measurable g.Measurable hf hg) (eapprox f n) -#align measurable.ennreal_induction Measurable.eNNReal_induction +#align measurable.ennreal_induction Measurable.ennreal_induction diff --git a/Mathbin/MeasureTheory/Function/SimpleFuncDense.lean b/Mathbin/MeasureTheory/Function/SimpleFuncDense.lean index 0c4ac3a644..af05d7d3d1 100644 --- a/Mathbin/MeasureTheory/Function/SimpleFuncDense.lean +++ b/Mathbin/MeasureTheory/Function/SimpleFuncDense.lean @@ -56,6 +56,7 @@ namespace SimpleFunc variable [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] +#print MeasureTheory.SimpleFunc.nearestPtInd /- /-- `nearest_pt_ind e N x` is the index `k` such that `e k` is the nearest point to `x` among the points `e 0`, ..., `e N`. If more than one point are at the same distance from `x`, then `nearest_pt_ind e N x` returns the least of their indexes. -/ @@ -68,24 +69,37 @@ noncomputable def nearestPtInd (e : ℕ → α) : ℕ → α →ₛ ℕ measurableSet_lt measurable_edist_right measurable_edist_right) (const α <| N + 1) (nearest_pt_ind N) #align measure_theory.simple_func.nearest_pt_ind MeasureTheory.SimpleFunc.nearestPtInd +-/ +#print MeasureTheory.SimpleFunc.nearestPt /- /-- `nearest_pt e N x` is the nearest point to `x` among the points `e 0`, ..., `e N`. If more than one point are at the same distance from `x`, then `nearest_pt e N x` returns the point with the least possible index. -/ noncomputable def nearestPt (e : ℕ → α) (N : ℕ) : α →ₛ α := (nearestPtInd e N).map e #align measure_theory.simple_func.nearest_pt MeasureTheory.SimpleFunc.nearestPt +-/ +#print MeasureTheory.SimpleFunc.nearestPtInd_zero /- @[simp] theorem nearestPtInd_zero (e : ℕ → α) : nearestPtInd e 0 = const α 0 := rfl #align measure_theory.simple_func.nearest_pt_ind_zero MeasureTheory.SimpleFunc.nearestPtInd_zero +-/ +#print MeasureTheory.SimpleFunc.nearestPt_zero /- @[simp] theorem nearestPt_zero (e : ℕ → α) : nearestPt e 0 = const α (e 0) := rfl #align measure_theory.simple_func.nearest_pt_zero MeasureTheory.SimpleFunc.nearestPt_zero +-/ +/- warning: measure_theory.simple_func.nearest_pt_ind_succ -> MeasureTheory.SimpleFunc.nearestPtInd_succ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] (e : Nat -> α) (N : Nat) (x : α), Eq.{1} Nat (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Nat) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Nat) => α -> Nat) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α Nat _inst_1) (MeasureTheory.SimpleFunc.nearestPtInd.{u1} α _inst_1 _inst_2 _inst_3 e (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) N (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) x) (ite.{1} Nat (forall (k : Nat), (LE.le.{0} Nat Nat.hasLe k N) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α 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(OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Nat) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α _inst_1 Nat) => α -> Nat) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α Nat _inst_1) (MeasureTheory.SimpleFunc.nearestPtInd.{u1} α _inst_1 _inst_2 _inst_3 e N) x)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] (e : Nat -> α) (N : Nat) (x : α), Eq.{1} Nat (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 Nat (MeasureTheory.SimpleFunc.nearestPtInd.{u1} α _inst_1 _inst_2 _inst_3 e (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) N (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x) (ite.{1} Nat (forall (k : Nat), (LE.le.{0} Nat instLENat k N) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) N (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e k) x))) (Nat.decidableBallLe N (fun (k : Nat) (H : LE.le.{0} Nat instLENat k N) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) N (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e k) x)) (fun (n : Nat) (h : LE.le.{0} Nat instLENat n N) => (fun (a : Nat) => Classical.propDecidable (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) N (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e a) x))) n)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) N (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α _inst_1 Nat (MeasureTheory.SimpleFunc.nearestPtInd.{u1} α _inst_1 _inst_2 _inst_3 e N) x)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.nearest_pt_ind_succ MeasureTheory.SimpleFunc.nearestPtInd_succₓ'. -/ theorem nearestPtInd_succ (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e (N + 1) x = if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else nearestPtInd e N x := @@ -95,6 +109,7 @@ theorem nearestPtInd_succ (e : ℕ → α) (N : ℕ) (x : α) : simp #align measure_theory.simple_func.nearest_pt_ind_succ MeasureTheory.SimpleFunc.nearestPtInd_succ +#print MeasureTheory.SimpleFunc.nearestPtInd_le /- theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N := by induction' N with N ihN; · simp @@ -102,7 +117,14 @@ theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x split_ifs exacts[le_rfl, ihN.trans N.le_succ] #align measure_theory.simple_func.nearest_pt_ind_le MeasureTheory.SimpleFunc.nearestPtInd_le +-/ +/- warning: measure_theory.simple_func.edist_nearest_pt_le -> MeasureTheory.SimpleFunc.edist_nearestPt_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] (e : Nat -> α) (x : α) {k : Nat} {N : Nat}, (LE.le.{0} Nat Nat.hasLe k N) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, u1} α _inst_1 α) (fun (_x : MeasureTheory.SimpleFunc.{u1, u1} α _inst_1 α) => α -> α) (MeasureTheory.SimpleFunc.instCoeFun.{u1, u1} α α _inst_1) (MeasureTheory.SimpleFunc.nearestPt.{u1} α _inst_1 _inst_2 _inst_3 e N) x) x) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) (e k) x)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] (e : Nat -> α) (x : α) {k : Nat} {N : Nat}, (LE.le.{0} Nat instLENat k N) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (MeasureTheory.SimpleFunc.toFun.{u1, u1} α _inst_1 α (MeasureTheory.SimpleFunc.nearestPt.{u1} α _inst_1 _inst_2 _inst_3 e N) x) x) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (e k) x)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.edist_nearest_pt_le MeasureTheory.SimpleFunc.edist_nearestPt_leₓ'. -/ theorem edist_nearestPt_le (e : ℕ → α) (x : α) {k N : ℕ} (hk : k ≤ N) : edist (nearestPt e N x) x ≤ edist (e k) x := by @@ -118,6 +140,7 @@ theorem edist_nearestPt_le (e : ℕ → α) (x : α) {k N : ℕ} (hk : k ≤ N) exacts[(ihN hlN).trans hxl, ihN (Nat.lt_succ_iff.1 hk)] #align measure_theory.simple_func.edist_nearest_pt_le MeasureTheory.SimpleFunc.edist_nearestPt_le +#print MeasureTheory.SimpleFunc.tendsto_nearestPt /- theorem tendsto_nearestPt {e : ℕ → α} {x : α} (hx : x ∈ closure (range e)) : Tendsto (fun N => nearestPt e N x) atTop (𝓝 x) := by @@ -126,9 +149,11 @@ theorem tendsto_nearestPt {e : ℕ → α} {x : α} (hx : x ∈ closure (range e rw [edist_comm] at hN exact ⟨N, trivial, fun n hn => (edist_nearest_pt_le e x hn).trans_lt hN⟩ #align measure_theory.simple_func.tendsto_nearest_pt MeasureTheory.SimpleFunc.tendsto_nearestPt +-/ variable [MeasurableSpace β] {f : β → α} +#print MeasureTheory.SimpleFunc.approxOn /- /-- Approximate a measurable function by a sequence of simple functions `F n` such that `F n x ∈ s`. -/ noncomputable def approxOn (f : β → α) (hf : Measurable f) (s : Set α) (y₀ : α) (h₀ : y₀ ∈ s) @@ -136,13 +161,17 @@ noncomputable def approxOn (f : β → α) (hf : Measurable f) (s : Set α) (y haveI : Nonempty s := ⟨⟨y₀, h₀⟩⟩ comp (nearest_pt (fun k => Nat.casesOn k y₀ (coe ∘ dense_seq s) : ℕ → α) n) f hf #align measure_theory.simple_func.approx_on MeasureTheory.SimpleFunc.approxOn +-/ +#print MeasureTheory.SimpleFunc.approxOn_zero /- @[simp] theorem approxOn_zero {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) : approxOn f hf s y₀ h₀ 0 x = y₀ := rfl #align measure_theory.simple_func.approx_on_zero MeasureTheory.SimpleFunc.approxOn_zero +-/ +#print MeasureTheory.SimpleFunc.approxOn_mem /- theorem approxOn_mem {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (n : ℕ) (x : β) : approxOn f hf s y₀ h₀ n x ∈ s := by @@ -151,14 +180,18 @@ theorem approxOn_mem {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α rintro (_ | n) exacts[h₀, Subtype.mem _] #align measure_theory.simple_func.approx_on_mem MeasureTheory.SimpleFunc.approxOn_mem +-/ +#print MeasureTheory.SimpleFunc.approxOn_comp /- @[simp] theorem approxOn_comp {γ : Type _} [MeasurableSpace γ] {f : β → α} (hf : Measurable f) {g : γ → β} (hg : Measurable g) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (n : ℕ) : approxOn (f ∘ g) (hf.comp hg) s y₀ h₀ n = (approxOn f hf s y₀ h₀ n).comp g hg := rfl #align measure_theory.simple_func.approx_on_comp MeasureTheory.SimpleFunc.approxOn_comp +-/ +#print MeasureTheory.SimpleFunc.tendsto_approxOn /- theorem tendsto_approxOn {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] {x : β} (hx : f x ∈ closure s) : Tendsto (fun n => approxOn f hf s y₀ h₀ n x) atTop (𝓝 <| f x) := @@ -172,7 +205,14 @@ theorem tendsto_approxOn {f : β → α} (hf : Measurable f) {s : Set α} {y₀ subset.trans (image_closure_subset_closure_image continuous_subtype_val) (subset_union_right _ _) #align measure_theory.simple_func.tendsto_approx_on MeasureTheory.SimpleFunc.tendsto_approxOn +-/ +/- warning: measure_theory.simple_func.edist_approx_on_mono -> MeasureTheory.SimpleFunc.edist_approxOn_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) {m : Nat} {n : Nat}, (LE.le.{0} Nat Nat.hasLe m n) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) (fun (_x : MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) => β -> α) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u1} β α _inst_4) (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x) (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) (fun (_x : MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) => β -> α) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u1} β α _inst_4) (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 m) x) (f x))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) {m : Nat} {n : Nat}, (LE.le.{0} Nat instLENat m n) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (MeasureTheory.SimpleFunc.toFun.{u2, u1} β _inst_4 α (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x) (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (MeasureTheory.SimpleFunc.toFun.{u2, u1} β _inst_4 α (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 m) x) (f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.edist_approx_on_mono MeasureTheory.SimpleFunc.edist_approxOn_monoₓ'. -/ theorem edist_approxOn_mono {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) {m n : ℕ} (h : m ≤ n) : edist (approxOn f hf s y₀ h₀ n x) (f x) ≤ edist (approxOn f hf s y₀ h₀ m x) (f x) := @@ -181,11 +221,23 @@ theorem edist_approxOn_mono {f : β → α} (hf : Measurable f) {s : Set α} {y exact edist_nearest_pt_le _ _ ((nearest_pt_ind_le _ _ _).trans h) #align measure_theory.simple_func.edist_approx_on_mono MeasureTheory.SimpleFunc.edist_approxOn_mono +/- warning: measure_theory.simple_func.edist_approx_on_le -> MeasureTheory.SimpleFunc.edist_approxOn_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) (n : Nat), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) (fun (_x : MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) => β -> α) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u1} β α _inst_4) (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x) (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) y₀ (f x)) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) (MeasureTheory.SimpleFunc.toFun.{u2, u1} β _inst_4 α (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x) (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) y₀ (f x)) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.edist_approx_on_le MeasureTheory.SimpleFunc.edist_approxOn_leₓ'. -/ theorem edist_approxOn_le {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) : edist (approxOn f hf s y₀ h₀ n x) (f x) ≤ edist y₀ (f x) := edist_approxOn_mono hf h₀ x (zero_le n) #align measure_theory.simple_func.edist_approx_on_le MeasureTheory.SimpleFunc.edist_approxOn_le +/- warning: measure_theory.simple_func.edist_approx_on_y0_le -> MeasureTheory.SimpleFunc.edist_approxOn_y0_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) (n : Nat), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) y₀ (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) (fun (_x : MeasureTheory.SimpleFunc.{u2, u1} β _inst_4 α) => β -> α) (MeasureTheory.SimpleFunc.instCoeFun.{u2, u1} β α _inst_4) (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) y₀ (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_2) y₀ (f x))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : PseudoEMetricSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)) _inst_1] [_inst_4 : MeasurableSpace.{u2} β] {f : β -> α} (hf : Measurable.{u2, u1} β α _inst_4 _inst_1 f) {s : Set.{u1} α} {y₀ : α} (h₀ : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y₀ s) [_inst_5 : TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (UniformSpace.toTopologicalSpace.{u1} α (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_2)))] (x : β) (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) y₀ (MeasureTheory.SimpleFunc.toFun.{u2, u1} β _inst_4 α (MeasureTheory.SimpleFunc.approxOn.{u1, u2} α β _inst_1 _inst_2 _inst_3 _inst_4 f hf s y₀ h₀ _inst_5 n) x)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) y₀ (f x)) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_2) y₀ (f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.edist_approx_on_y0_le MeasureTheory.SimpleFunc.edist_approxOn_y0_leₓ'. -/ theorem edist_approxOn_y0_le {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) : edist y₀ (approxOn f hf s y₀ h₀ n x) ≤ edist y₀ (f x) + edist y₀ (f x) := diff --git a/Mathbin/MeasureTheory/Integral/Lebesgue.lean b/Mathbin/MeasureTheory/Integral/Lebesgue.lean index 4b853f1a88..5ba14476a6 100644 --- a/Mathbin/MeasureTheory/Integral/Lebesgue.lean +++ b/Mathbin/MeasureTheory/Integral/Lebesgue.lean @@ -50,6 +50,7 @@ variable {α : Type _} {mα : MeasurableSpace α} {a : α} {s : Set α} include mα +#print MeasureTheory.restrict_dirac' /- -- todo after the port: move to measure_theory/measure/measure_space theorem restrict_dirac' (hs : MeasurableSet s) [Decidable (a ∈ s)] : (Measure.dirac a).restrict s = if a ∈ s then Measure.dirac a else 0 := @@ -65,7 +66,9 @@ theorem restrict_dirac' (hs : MeasurableSet s) [Decidable (a ∈ s)] : · simp only [measure.dirac_apply' _ ht, Set.indicator_apply, hat, if_false] · simp only [has, and_false_iff, if_false, measure.coe_zero, Pi.zero_apply] #align measure_theory.restrict_dirac' MeasureTheory.restrict_dirac' +-/ +#print MeasureTheory.restrict_dirac /- -- todo after the port: move to measure_theory/measure/measure_space theorem restrict_dirac [MeasurableSingletonClass α] [Decidable (a ∈ s)] : (Measure.dirac a).restrict s = if a ∈ s then Measure.dirac a else 0 := @@ -81,6 +84,7 @@ theorem restrict_dirac [MeasurableSingletonClass α] [Decidable (a ∈ s)] : · simp only [measure.dirac_apply' _ ht, Set.indicator_apply, hat, if_false] · simp only [has, and_false_iff, if_false, measure.coe_zero, Pi.zero_apply] #align measure_theory.restrict_dirac MeasureTheory.restrict_dirac +-/ end MoveThis @@ -95,10 +99,12 @@ open SimpleFunc variable {m : MeasurableSpace α} {μ ν : Measure α} +#print MeasureTheory.lintegral /- /-- The **lower Lebesgue integral** of a function `f` with respect to a measure `μ`. -/ irreducible_def lintegral {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ := ⨆ (g : α →ₛ ℝ≥0∞) (hf : ⇑g ≤ f), g.lintegral μ #align measure_theory.lintegral MeasureTheory.lintegral +-/ /-! In the notation for integrals, an expression like `∫⁻ x, g ‖x‖ ∂μ` will not be parsed correctly, and needs parentheses. We do not set the binding power of `r` to `0`, because then @@ -117,13 +123,21 @@ notation3"∫⁻ "(...)" in "s", "r:(scoped f => f)" ∂"μ => lintegral (Measur -- mathport name: «expr∫⁻ in , » notation3"∫⁻ "(...)" in "s", "r:(scoped f => lintegral Measure.restrict volume s f) => r +#print MeasureTheory.SimpleFunc.lintegral_eq_lintegral /- theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) : (∫⁻ a, f a ∂μ) = f.lintegral μ := by rw [lintegral] exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl) (le_iSup₂_of_le f le_rfl le_rfl) #align measure_theory.simple_func.lintegral_eq_lintegral MeasureTheory.SimpleFunc.lintegral_eq_lintegral +-/ +/- warning: measure_theory.lintegral_mono' -> MeasureTheory.lintegral_mono' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {{ν : MeasureTheory.Measure.{u1} α m}}, (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toHasLe.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) μ ν) -> (forall {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) f g) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m ν (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {{ν : MeasureTheory.Measure.{u1} α m}}, (LE.le.{u1} (MeasureTheory.Measure.{u1} α m) (Preorder.toLE.{u1} (MeasureTheory.Measure.{u1} α m) (PartialOrder.toPreorder.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instPartialOrder.{u1} α m))) μ ν) -> (forall {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) f g) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m ν (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono' MeasureTheory.lintegral_mono'ₓ'. -/ @[mono] theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : (∫⁻ a, f a ∂μ) ≤ ∫⁻ a, g a ∂ν := @@ -132,14 +146,32 @@ theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν exact iSup_mono fun φ => iSup_mono' fun hφ => ⟨le_trans hφ hfg, lintegral_mono (le_refl φ) hμν⟩ #align measure_theory.lintegral_mono' MeasureTheory.lintegral_mono' +/- warning: measure_theory.lintegral_mono -> MeasureTheory.lintegral_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) f g) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {{f : α -> ENNReal}} {{g : α -> ENNReal}}, (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) f g) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono MeasureTheory.lintegral_monoₓ'. -/ theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : (∫⁻ a, f a ∂μ) ≤ ∫⁻ a, g a ∂μ := lintegral_mono' (le_refl μ) hfg #align measure_theory.lintegral_mono MeasureTheory.lintegral_mono -theorem lintegral_mono_nNReal {f g : α → ℝ≥0} (h : f ≤ g) : (∫⁻ a, f a ∂μ) ≤ ∫⁻ a, g a ∂μ := +/- warning: measure_theory.lintegral_mono_nnreal -> MeasureTheory.lintegral_mono_nnreal is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> NNReal} {g : α -> NNReal}, (LE.le.{u1} (α -> NNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => NNReal) (fun (i : α) => Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring))))) f g) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> NNReal} {g : α -> NNReal}, (LE.le.{u1} (α -> NNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => NNReal) (fun (i : α) => Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring)))) f g) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => ENNReal.some (f a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => ENNReal.some (g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nnrealₓ'. -/ +theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (h : f ≤ g) : (∫⁻ a, f a ∂μ) ≤ ∫⁻ a, g a ∂μ := lintegral_mono fun a => ENNReal.coe_le_coe.2 (h a) -#align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nNReal - +#align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nnreal + +/- warning: measure_theory.supr_lintegral_measurable_le_eq_lintegral -> MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal), Eq.{1} ENNReal (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (α -> ENNReal) (fun (g : α -> ENNReal) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) (fun (g_meas : Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) g f) (fun (hg : LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) g f) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal), Eq.{1} ENNReal (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (α -> ENNReal) (fun (g : α -> ENNReal) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) (fun (g_meas : Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) g f) (fun (hg : LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) g f) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.supr_lintegral_measurable_le_eq_lintegral MeasureTheory.iSup_lintegral_measurable_le_eq_lintegralₓ'. -/ theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) : (⨆ (g : α → ℝ≥0∞) (g_meas : Measurable g) (hg : g ≤ f), ∫⁻ a, g a ∂μ) = ∫⁻ a, f a ∂μ := by @@ -150,49 +182,107 @@ theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) : exact le_of_eq (i.lintegral_eq_lintegral _).symm #align measure_theory.supr_lintegral_measurable_le_eq_lintegral MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral +/- warning: measure_theory.lintegral_mono_set -> MeasureTheory.lintegral_mono_set is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {s : Set.{u1} α} {t : Set.{u1} α} {f : α -> ENNReal}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t) (fun (x : α) => f x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {s : Set.{u1} α} {t : Set.{u1} α} {f : α -> ENNReal}, (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s t) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t) (fun (x : α) => f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono_set MeasureTheory.lintegral_mono_setₓ'. -/ theorem lintegral_mono_set {m : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞} (hst : s ⊆ t) : (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in t, f x ∂μ := lintegral_mono' (Measure.restrict_mono hst (le_refl μ)) (le_refl f) #align measure_theory.lintegral_mono_set MeasureTheory.lintegral_mono_set +/- warning: measure_theory.lintegral_mono_set' -> MeasureTheory.lintegral_mono_set' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {s : Set.{u1} α} {t : Set.{u1} α} {f : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α Prop Prop.le (MeasureTheory.Measure.ae.{u1} α m μ) s t) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t) (fun (x : α) => f x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {{μ : MeasureTheory.Measure.{u1} α m}} {s : Set.{u1} α} {t : Set.{u1} α} {f : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α Prop Prop.le (MeasureTheory.Measure.ae.{u1} α m μ) s t) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ t) (fun (x : α) => f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono_set' MeasureTheory.lintegral_mono_set'ₓ'. -/ theorem lintegral_mono_set' {m : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞} (hst : s ≤ᵐ[μ] t) : (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in t, f x ∂μ := lintegral_mono' (Measure.restrict_mono' hst (le_refl μ)) (le_refl f) #align measure_theory.lintegral_mono_set' MeasureTheory.lintegral_mono_set' +/- warning: measure_theory.monotone_lintegral -> MeasureTheory.monotone_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m), Monotone.{u1, 0} (α -> ENNReal) ENNReal (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.lintegral.{u1} α m μ) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m), Monotone.{u1, 0} (α -> ENNReal) ENNReal (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.lintegral.{u1} α m μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.monotone_lintegral MeasureTheory.monotone_lintegralₓ'. -/ theorem monotone_lintegral {m : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) := lintegral_mono #align measure_theory.monotone_lintegral MeasureTheory.monotone_lintegral +/- warning: measure_theory.lintegral_const -> MeasureTheory.lintegral_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const MeasureTheory.lintegral_constₓ'. -/ @[simp] theorem lintegral_const (c : ℝ≥0∞) : (∫⁻ a, c ∂μ) = c * μ univ := by rw [← simple_func.const_lintegral, ← simple_func.lintegral_eq_lintegral, simple_func.coe_const] #align measure_theory.lintegral_const MeasureTheory.lintegral_const +/- warning: measure_theory.lintegral_zero -> MeasureTheory.lintegral_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_zero MeasureTheory.lintegral_zeroₓ'. -/ theorem lintegral_zero : (∫⁻ a : α, 0 ∂μ) = 0 := by simp #align measure_theory.lintegral_zero MeasureTheory.lintegral_zero +/- warning: measure_theory.lintegral_zero_fun -> MeasureTheory.lintegral_zero_fun is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (OfNat.ofNat.{u1} (α -> ENNReal) 0 (OfNat.mk.{u1} (α -> ENNReal) 0 (Zero.zero.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => ENNReal.hasZero)))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m}, Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (OfNat.ofNat.{u1} (α -> ENNReal) 0 (Zero.toOfNat0.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.5462 : α) => ENNReal) (fun (i : α) => instENNRealZero))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_zero_fun MeasureTheory.lintegral_zero_funₓ'. -/ theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 := lintegral_zero #align measure_theory.lintegral_zero_fun MeasureTheory.lintegral_zero_fun +#print MeasureTheory.lintegral_one /- @[simp] theorem lintegral_one : (∫⁻ a, (1 : ℝ≥0∞) ∂μ) = μ univ := by rw [lintegral_const, one_mul] #align measure_theory.lintegral_one MeasureTheory.lintegral_one +-/ +/- warning: measure_theory.set_lintegral_const -> MeasureTheory.set_lintegral_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Set.{u1} α) (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Set.{u1} α) (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s)) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_const MeasureTheory.set_lintegral_constₓ'. -/ theorem set_lintegral_const (s : Set α) (c : ℝ≥0∞) : (∫⁻ a in s, c ∂μ) = c * μ s := by rw [lintegral_const, measure.restrict_apply_univ] #align measure_theory.set_lintegral_const MeasureTheory.set_lintegral_const +#print MeasureTheory.set_lintegral_one /- theorem set_lintegral_one (s) : (∫⁻ a in s, 1 ∂μ) = μ s := by rw [set_lintegral_const, one_mul] #align measure_theory.set_lintegral_one MeasureTheory.set_lintegral_one +-/ +/- warning: measure_theory.set_lintegral_const_lt_top -> MeasureTheory.set_lintegral_const_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasureTheory.FiniteMeasure.{u1} α m μ] (s : Set.{u1} α) {c : ENNReal}, (Ne.{1} ENNReal c (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => c)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasureTheory.FiniteMeasure.{u1} α m μ] (s : Set.{u1} α) {c : ENNReal}, (Ne.{1} ENNReal c (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => c)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_const_lt_top MeasureTheory.set_lintegral_const_lt_topₓ'. -/ theorem set_lintegral_const_lt_top [FiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) : (∫⁻ a in s, c ∂μ) < ∞ := by rw [lintegral_const] exact ENNReal.mul_lt_top hc (measure_ne_top (μ.restrict s) univ) #align measure_theory.set_lintegral_const_lt_top MeasureTheory.set_lintegral_const_lt_top +/- warning: measure_theory.lintegral_const_lt_top -> MeasureTheory.lintegral_const_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasureTheory.FiniteMeasure.{u1} α m μ] {c : ENNReal}, (Ne.{1} ENNReal c (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => c)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasureTheory.FiniteMeasure.{u1} α m μ] {c : ENNReal}, (Ne.{1} ENNReal c (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => c)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const_lt_top MeasureTheory.lintegral_const_lt_topₓ'. -/ theorem lintegral_const_lt_top [FiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : (∫⁻ a, c ∂μ) < ∞ := by simpa only [measure.restrict_univ] using set_lintegral_const_lt_top univ hc #align measure_theory.lintegral_const_lt_top MeasureTheory.lintegral_const_lt_top @@ -201,6 +291,12 @@ section variable (μ) +/- warning: measure_theory.exists_measurable_le_lintegral_eq -> MeasureTheory.exists_measurable_le_lintegral_eq is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) (f : α -> ENNReal), Exists.{succ u1} (α -> ENNReal) (fun (g : α -> ENNReal) => And (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) (And (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) g f) (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) (f : α -> ENNReal), Exists.{succ u1} (α -> ENNReal) (fun (g : α -> ENNReal) => And (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) (And (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) g f) (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_measurable_le_lintegral_eq MeasureTheory.exists_measurable_le_lintegral_eqₓ'. -/ /-- For any function `f : α → ℝ≥0∞`, there exists a measurable function `g ≤ f` with the same integral. -/ theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) : @@ -224,10 +320,16 @@ theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) : end +/- warning: measure_theory.lintegral_eq_nnreal -> MeasureTheory.lintegral_eq_nnreal is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (φ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (forall (x : α), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) φ x)) (f x)) (fun (hf : forall (x : α), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) φ x)) (f x)) => MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal m ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe)))) φ) μ))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (φ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (forall (x : α), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal φ x)) (f x)) (fun (hf : forall (x : α), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal φ x)) (f x)) => MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal m ENNReal.some φ) μ))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nnrealₓ'. -/ /-- `∫⁻ a in s, f a ∂μ` is defined as the supremum of integrals of simple functions `φ : α →ₛ ℝ≥0∞` such that `φ ≤ f`. This lemma says that it suffices to take functions `φ : α →ₛ ℝ≥0`. -/ -theorem lintegral_eq_nNReal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) : +theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) : (∫⁻ a, f a ∂μ) = ⨆ (φ : α →ₛ ℝ≥0) (hf : ∀ x, ↑(φ x) ≤ f x), (φ.map (coe : ℝ≥0 → ℝ≥0∞)).lintegral μ := by @@ -251,8 +353,14 @@ theorem lintegral_eq_nNReal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ refine' ⟨indicator_le fun x hx => le_trans _ (hφ _), hn⟩ simp only [mem_preimage, mem_singleton_iff] at hx simp only [hx, le_top] -#align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nNReal - +#align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nnreal + +/- warning: measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos -> MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (φ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) φ x)) (f x)) (forall (ψ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal), (forall (x : α), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) ψ x)) (f x)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal m ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe)))) (HSub.hSub.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (instHSub.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.instSub.{u1, 0} α NNReal m NNReal.hasSub)) ψ φ)) μ) ε))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (φ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal φ x)) (f x)) (forall (ψ : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal), (forall (x : α), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal ψ x)) (f x)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.map.{u1, 0, 0} α NNReal ENNReal m ENNReal.some (HSub.hSub.{u1, u1, u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (instHSub.{u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (MeasureTheory.SimpleFunc.instSub.{u1, 0} α NNReal m NNReal.instSubNNReal)) ψ φ)) μ) ε))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_posₓ'. -/ theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ φ : α →ₛ ℝ≥0, @@ -272,6 +380,12 @@ theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} simp only [add_apply, sub_apply, add_tsub_eq_max] #align measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos +/- warning: measure_theory.supr_lintegral_le -> MeasureTheory.iSup_lintegral_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} (f : ι -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => f i a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} (f : ι -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => f i a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.supr_lintegral_le MeasureTheory.iSup_lintegral_leₓ'. -/ theorem iSup_lintegral_le {ι : Sort _} (f : ι → α → ℝ≥0∞) : (⨆ i, ∫⁻ a, f i a ∂μ) ≤ ∫⁻ a, ⨆ i, f i a ∂μ := by @@ -279,16 +393,28 @@ theorem iSup_lintegral_le {ι : Sort _} (f : ι → α → ℝ≥0∞) : exact (monotone_lintegral μ).le_map_iSup #align measure_theory.supr_lintegral_le MeasureTheory.iSup_lintegral_le +/- warning: measure_theory.supr₂_lintegral_le -> MeasureTheory.iSup₂_lintegral_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} {ι' : ι -> Sort.{u3}} (f : forall (i : ι), (ι' i) -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => iSup.{0, u3} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (ι' i) (fun (j : ι' i) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i j a)))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => iSup.{0, u3} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (ι' i) (fun (j : ι' i) => f i j a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u3}} {ι' : ι -> Sort.{u2}} (f : forall (i : ι), (ι' i) -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (iSup.{0, u3} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (ι' i) (fun (j : ι' i) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i j a)))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, u3} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => iSup.{0, u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (ι' i) (fun (j : ι' i) => f i j a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.supr₂_lintegral_le MeasureTheory.iSup₂_lintegral_leₓ'. -/ /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/ /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/ -theorem supr₂_lintegral_le {ι : Sort _} {ι' : ι → Sort _} (f : ∀ i, ι' i → α → ℝ≥0∞) : +theorem iSup₂_lintegral_le {ι : Sort _} {ι' : ι → Sort _} (f : ∀ i, ι' i → α → ℝ≥0∞) : (⨆ (i) (j), ∫⁻ a, f i j a ∂μ) ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ := by convert(monotone_lintegral μ).le_map_iSup₂ f ext1 a simp only [iSup_apply] -#align measure_theory.supr₂_lintegral_le MeasureTheory.supr₂_lintegral_le - +#align measure_theory.supr₂_lintegral_le MeasureTheory.iSup₂_lintegral_le + +/- warning: measure_theory.le_infi_lintegral -> MeasureTheory.le_iInf_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} (f : ι -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => f i a))) (iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} (f : ι -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => f i a))) (iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.le_infi_lintegral MeasureTheory.le_iInf_lintegralₓ'. -/ theorem le_iInf_lintegral {ι : Sort _} (f : ι → α → ℝ≥0∞) : (∫⁻ a, ⨅ i, f i a ∂μ) ≤ ⨅ i, ∫⁻ a, f i a ∂μ := by @@ -296,14 +422,26 @@ theorem le_iInf_lintegral {ι : Sort _} (f : ι → α → ℝ≥0∞) : exact (monotone_lintegral μ).map_iInf_le #align measure_theory.le_infi_lintegral MeasureTheory.le_iInf_lintegral -theorem le_infi₂_lintegral {ι : Sort _} {ι' : ι → Sort _} (f : ∀ i, ι' i → α → ℝ≥0∞) : +/- warning: measure_theory.le_infi₂_lintegral -> MeasureTheory.le_iInf₂_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u2}} {ι' : ι -> Sort.{u3}} (f : forall (i : ι), (ι' i) -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => iInf.{0, u3} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (ι' i) (fun (h : ι' i) => f i h a)))) (iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) ι (fun (i : ι) => iInf.{0, u3} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (ι' i) (fun (h : ι' i) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i h a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Sort.{u3}} {ι' : ι -> Sort.{u2}} (f : forall (i : ι), (ι' i) -> α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, u3} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (ι' i) (fun (h : ι' i) => f i h a)))) (iInf.{0, u3} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) ι (fun (i : ι) => iInf.{0, u2} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (ι' i) (fun (h : ι' i) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i h a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.le_infi₂_lintegral MeasureTheory.le_iInf₂_lintegralₓ'. -/ +theorem le_iInf₂_lintegral {ι : Sort _} {ι' : ι → Sort _} (f : ∀ i, ι' i → α → ℝ≥0∞) : (∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ) ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ := by convert(monotone_lintegral μ).map_iInf₂_le f ext1 a simp only [iInf_apply] -#align measure_theory.le_infi₂_lintegral MeasureTheory.le_infi₂_lintegral - +#align measure_theory.le_infi₂_lintegral MeasureTheory.le_iInf₂_lintegral + +/- warning: measure_theory.lintegral_mono_ae -> MeasureTheory.lintegral_mono_ae is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f a) (g a)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f a) (g a)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mono_ae MeasureTheory.lintegral_mono_aeₓ'. -/ theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) : (∫⁻ a, f a ∂μ) ≤ ∫⁻ a, g a ∂μ := by @@ -319,28 +457,47 @@ theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ exact (hnt hat).elim #align measure_theory.lintegral_mono_ae MeasureTheory.lintegral_mono_ae +/- warning: measure_theory.set_lintegral_mono_ae -> MeasureTheory.set_lintegral_mono_ae is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_mono_ae MeasureTheory.set_lintegral_mono_aeₓ'. -/ theorem set_lintegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ := lintegral_mono_ae <| (ae_restrict_iff <| measurableSet_le hf hg).2 hfg #align measure_theory.set_lintegral_mono_ae MeasureTheory.set_lintegral_mono_ae +/- warning: measure_theory.set_lintegral_mono -> MeasureTheory.set_lintegral_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_mono MeasureTheory.set_lintegral_monoₓ'. -/ theorem set_lintegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ x ∈ s, f x ≤ g x) : (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ := set_lintegral_mono_ae hf hg (ae_of_all _ hfg) #align measure_theory.set_lintegral_mono MeasureTheory.set_lintegral_mono +#print MeasureTheory.lintegral_congr_ae /- theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : (∫⁻ a, f a ∂μ) = ∫⁻ a, g a ∂μ := le_antisymm (lintegral_mono_ae <| h.le) (lintegral_mono_ae <| h.symm.le) #align measure_theory.lintegral_congr_ae MeasureTheory.lintegral_congr_ae +-/ +#print MeasureTheory.lintegral_congr /- theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : (∫⁻ a, f a ∂μ) = ∫⁻ a, g a ∂μ := by simp only [h] #align measure_theory.lintegral_congr MeasureTheory.lintegral_congr +-/ +#print MeasureTheory.set_lintegral_congr /- theorem set_lintegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) : (∫⁻ x in s, f x ∂μ) = ∫⁻ x in t, f x ∂μ := by rw [measure.restrict_congr_set h] #align measure_theory.set_lintegral_congr MeasureTheory.set_lintegral_congr +-/ +#print MeasureTheory.set_lintegral_congr_fun /- theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s) (hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ := by @@ -348,7 +505,14 @@ theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : Mea rw [eventually_eq] rwa [ae_restrict_iff' hs] #align measure_theory.set_lintegral_congr_fun MeasureTheory.set_lintegral_congr_fun +-/ +/- warning: measure_theory.lintegral_of_real_le_lintegral_nnnorm -> MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> Real), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> Real), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_of_real_le_lintegral_nnnorm MeasureTheory.lintegral_ofReal_le_lintegral_nnnormₓ'. -/ theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) : (∫⁻ x, ENNReal.ofReal (f x) ∂μ) ≤ ∫⁻ x, ‖f x‖₊ ∂μ := by @@ -358,6 +522,12 @@ theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) : exact le_abs_self (f x) #align measure_theory.lintegral_of_real_le_lintegral_nnnorm MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm +/- warning: measure_theory.lintegral_nnnorm_eq_of_ae_nonneg -> MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (Filter.EventuallyLE.{u1, 0} α Real Real.hasLe (MeasureTheory.Measure.ae.{u1} α m μ) (OfNat.ofNat.{u1} (α -> Real) 0 (OfNat.mk.{u1} (α -> Real) 0 (Zero.zero.{u1} (α -> Real) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => Real) (fun (i : α) => Real.hasZero))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (Filter.EventuallyLE.{u1, 0} α Real Real.instLEReal (MeasureTheory.Measure.ae.{u1} α m μ) (OfNat.ofNat.{u1} (α -> Real) 0 (Zero.toOfNat0.{u1} (α -> Real) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.21854 : α) => Real) (fun (i : α) => Real.instZeroReal)))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_nnnorm_eq_of_ae_nonneg MeasureTheory.lintegral_nnnorm_eq_of_ae_nonnegₓ'. -/ theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) : (∫⁻ x, ‖f x‖₊ ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by @@ -366,11 +536,23 @@ theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[ rw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx] #align measure_theory.lintegral_nnnorm_eq_of_ae_nonneg MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg +/- warning: measure_theory.lintegral_nnnorm_eq_of_nonneg -> MeasureTheory.lintegral_nnnorm_eq_of_nonneg is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (LE.le.{u1} (α -> Real) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => Real) (fun (i : α) => Real.hasLe)) (OfNat.ofNat.{u1} (α -> Real) 0 (OfNat.mk.{u1} (α -> Real) 0 (Zero.zero.{u1} (α -> Real) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => Real) (fun (i : α) => Real.hasZero))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> Real}, (LE.le.{u1} (α -> Real) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => Real) (fun (i : α) => Real.instLEReal)) (OfNat.ofNat.{u1} (α -> Real) 0 (Zero.toOfNat0.{u1} (α -> Real) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.9034 : α) => Real) (fun (i : α) => Real.instZeroReal)))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Real Real.normedAddCommGroup))) (f x)))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.ofReal (f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_nnnorm_eq_of_nonneg MeasureTheory.lintegral_nnnorm_eq_of_nonnegₓ'. -/ theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) : (∫⁻ x, ‖f x‖₊ ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := lintegral_nnnorm_eq_of_ae_nonneg (Filter.eventually_of_forall h_nonneg) #align measure_theory.lintegral_nnnorm_eq_of_nonneg MeasureTheory.lintegral_nnnorm_eq_of_nonneg +/- warning: measure_theory.lintegral_supr -> MeasureTheory.lintegral_iSup is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (Monotone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (Monotone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_supr MeasureTheory.lintegral_iSupₓ'. -/ /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([1]) } -/ /-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. @@ -453,9 +635,15 @@ theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable ( #align measure_theory.lintegral_supr MeasureTheory.lintegral_iSup +/- warning: measure_theory.lintegral_supr' -> MeasureTheory.lintegral_iSup' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (Filter.Eventually.{u1} α (fun (x : α) => Monotone.{0, 0} Nat ENNReal (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (fun (n : Nat) => f n x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (Filter.Eventually.{u1} α (fun (x : α) => Monotone.{0, 0} Nat ENNReal (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (fun (n : Nat) => f n x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_supr' MeasureTheory.lintegral_iSup'ₓ'. -/ /-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with ae_measurable functions. -/ -theorem lintegral_supr' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) +theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : (∫⁻ a, ⨆ n, f n a ∂μ) = ⨆ n, ∫⁻ a, f n a ∂μ := by simp_rw [← iSup_apply] @@ -473,8 +661,14 @@ theorem lintegral_supr' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurabl rw [@lintegral_supr _ _ μ _ (aeSeq.measurable hf p) h_ae_seq_mono] congr exact funext fun n => lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n) -#align measure_theory.lintegral_supr' MeasureTheory.lintegral_supr' - +#align measure_theory.lintegral_supr' MeasureTheory.lintegral_iSup' + +/- warning: measure_theory.lintegral_tendsto_of_tendsto_of_monotone -> MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal} {F : α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (Filter.Eventually.{u1} α (fun (x : α) => Monotone.{0, 0} Nat ENNReal (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (fun (n : Nat) => f n x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => f n x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (F x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f n x)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => F x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal} {F : α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (Filter.Eventually.{u1} α (fun (x : α) => Monotone.{0, 0} Nat ENNReal (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (fun (n : Nat) => f n x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Eventually.{u1} α (fun (x : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => f n x) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (F x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f n x)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => F x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_tendsto_of_tendsto_of_monotone MeasureTheory.lintegral_tendsto_of_tendsto_of_monotoneₓ'. -/ /-- Monotone convergence theorem expressed with limits -/ theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) @@ -493,6 +687,12 @@ theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} (tendsto_atTop_iSup hx_mono) #align measure_theory.lintegral_tendsto_of_tendsto_of_monotone MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone +/- warning: measure_theory.lintegral_eq_supr_eapprox_lintegral -> MeasureTheory.lintegral_eq_iSup_eapprox_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.eapprox.{u1} α m f n) μ))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.SimpleFunc.lintegral.{u1} α m (MeasureTheory.SimpleFunc.eapprox.{u1} α m f n) μ))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_eq_supr_eapprox_lintegral MeasureTheory.lintegral_eq_iSup_eapprox_lintegralₓ'. -/ theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f a ∂μ) = ⨆ n, (eapprox f n).lintegral μ := calc @@ -509,6 +709,12 @@ theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measur #align measure_theory.lintegral_eq_supr_eapprox_lintegral MeasureTheory.lintegral_eq_iSup_eapprox_lintegral +/- warning: measure_theory.exists_pos_set_lintegral_lt_of_measure_lt -> MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{1} ENNReal (fun (δ : ENNReal) => Exists.{0} (GT.gt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) δ (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (fun (H : GT.gt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) δ (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) => forall (s : Set.{u1} α), (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) δ) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) ε))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{1} ENNReal (fun (δ : ENNReal) => And (GT.gt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) δ (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (forall (s : Set.{u1} α), (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) δ) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) ε))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_pos_set_lintegral_lt_of_measure_lt MeasureTheory.exists_pos_set_lintegral_lt_of_measure_ltₓ'. -/ /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends to zero as `μ s` tends to zero. This lemma states states this fact in terms of `ε` and `δ`. -/ theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} @@ -545,6 +751,12 @@ theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ( #align measure_theory.exists_pos_set_lintegral_lt_of_measure_lt MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt +/- warning: measure_theory.tendsto_set_lintegral_zero -> MeasureTheory.tendsto_set_lintegral_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Type.{u2}} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {l : Filter.{u2} ι} {s : ι -> (Set.{u1} α)}, (Filter.Tendsto.{u2, 0} ι ENNReal (Function.comp.{succ u2, succ u1, 1} ι (Set.{u1} α) ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ) s) l (nhds.{0} ENNReal ENNReal.topologicalSpace (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))))) -> (Filter.Tendsto.{u2, 0} ι ENNReal (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (x : α) => f x)) l (nhds.{0} ENNReal ENNReal.topologicalSpace (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Type.{u2}} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {l : Filter.{u2} ι} {s : ι -> (Set.{u1} α)}, (Filter.Tendsto.{u2, 0} ι ENNReal (Function.comp.{succ u2, succ u1, 1} ι (Set.{u1} α) ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ)) s) l (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)))) -> (Filter.Tendsto.{u2, 0} ι ENNReal (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (x : α) => f x)) l (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_set_lintegral_zero MeasureTheory.tendsto_set_lintegral_zeroₓ'. -/ /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends to zero as `μ s` tends to zero. -/ theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : (∫⁻ x, f x ∂μ) ≠ ∞) {l : Filter ι} @@ -557,6 +769,12 @@ theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : (∫⁻ x, exact (hl δ δ0).mono fun i => hδ _ #align measure_theory.tendsto_set_lintegral_zero MeasureTheory.tendsto_set_lintegral_zero +/- warning: measure_theory.le_lintegral_add -> MeasureTheory.le_lintegral_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.le_lintegral_add MeasureTheory.le_lintegral_addₓ'. -/ /-- The sum of the lower Lebesgue integrals of two functions is less than or equal to the integral of their sum. The other inequality needs one of these functions to be (a.e.-)measurable. -/ theorem le_lintegral_add (f g : α → ℝ≥0∞) : ((∫⁻ a, f a ∂μ) + ∫⁻ a, g a ∂μ) ≤ ∫⁻ a, f a + g a ∂μ := @@ -566,6 +784,12 @@ theorem le_lintegral_add (f g : α → ℝ≥0∞) : ((∫⁻ a, f a ∂μ) + exact le_iSup₂_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge #align measure_theory.le_lintegral_add MeasureTheory.le_lintegral_add +/- warning: measure_theory.lintegral_add_aux -> MeasureTheory.lintegral_add_aux is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_aux MeasureTheory.lintegral_add_auxₓ'. -/ -- Use stronger lemmas `lintegral_add_left`/`lintegral_add_right` instead theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) : (∫⁻ a, f a + g a ∂μ) = (∫⁻ a, f a ∂μ) + ∫⁻ a, g a ∂μ := @@ -600,6 +824,12 @@ theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Me #align measure_theory.lintegral_add_aux MeasureTheory.lintegral_add_aux +/- warning: measure_theory.lintegral_add_left -> MeasureTheory.lintegral_add_left is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_left MeasureTheory.lintegral_add_leftₓ'. -/ /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of `f + g` equals the sum of integrals. This lemma assumes that `f` is integrable, see also `measure_theory.lintegral_add_right` and primed versions of these lemmas. -/ @@ -618,17 +848,35 @@ theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α #align measure_theory.lintegral_add_left MeasureTheory.lintegral_add_left +/- warning: measure_theory.lintegral_add_left' -> MeasureTheory.lintegral_add_left' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_left' MeasureTheory.lintegral_add_left'ₓ'. -/ theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) : (∫⁻ a, f a + g a ∂μ) = (∫⁻ a, f a ∂μ) + ∫⁻ a, g a ∂μ := by rw [lintegral_congr_ae hf.ae_eq_mk, ← lintegral_add_left hf.measurable_mk, lintegral_congr_ae (hf.ae_eq_mk.add (ae_eq_refl g))] #align measure_theory.lintegral_add_left' MeasureTheory.lintegral_add_left' +/- warning: measure_theory.lintegral_add_right' -> MeasureTheory.lintegral_add_right' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_right' MeasureTheory.lintegral_add_right'ₓ'. -/ theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) : (∫⁻ a, f a + g a ∂μ) = (∫⁻ a, f a ∂μ) + ∫⁻ a, g a ∂μ := by simpa only [add_comm] using lintegral_add_left' hg f #align measure_theory.lintegral_add_right' MeasureTheory.lintegral_add_right' +/- warning: measure_theory.lintegral_add_right -> MeasureTheory.lintegral_add_right is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f a) (g a))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_right MeasureTheory.lintegral_add_rightₓ'. -/ /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of `f + g` equals the sum of integrals. This lemma assumes that `g` is integrable, see also `measure_theory.lintegral_add_left` and primed versions of these lemmas. -/ @@ -638,11 +886,23 @@ theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg lintegral_add_right' f hg.AEMeasurable #align measure_theory.lintegral_add_right MeasureTheory.lintegral_add_right +/- warning: measure_theory.lintegral_smul_measure -> MeasureTheory.lintegral_smul_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) m) c μ) (fun (a : α) => f a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (c : ENNReal) (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m)) c μ) (fun (a : α) => f a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_smul_measure MeasureTheory.lintegral_smul_measureₓ'. -/ @[simp] theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂c • μ) = c * ∫⁻ a, f a ∂μ := by simp only [lintegral, iSup_subtype', simple_func.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul] #align measure_theory.lintegral_smul_measure MeasureTheory.lintegral_smul_measure +/- warning: measure_theory.lintegral_sum_measure -> MeasureTheory.lintegral_sum_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {ι : Type.{u2}} (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.sum.{u1, u2} α ι m μ) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace ι (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (μ i) (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u2}} {m : MeasurableSpace.{u2} α} {ι : Type.{u1}} (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u2} α m)), Eq.{1} ENNReal (MeasureTheory.lintegral.{u2} α m (MeasureTheory.Measure.sum.{u2, u1} α ι m μ) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal ι (fun (i : ι) => MeasureTheory.lintegral.{u2} α m (μ i) (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_sum_measure MeasureTheory.lintegral_sum_measureₓ'. -/ @[simp] theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) : (∫⁻ a, f a ∂Measure.sum μ) = ∑' i, ∫⁻ a, f a ∂μ i := @@ -662,17 +922,35 @@ theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0 (Finset.sum_le_sum fun j hj => simple_func.lintegral_mono le_sup_right le_rfl)⟩ #align measure_theory.lintegral_sum_measure MeasureTheory.lintegral_sum_measure +/- warning: measure_theory.has_sum_lintegral_measure -> MeasureTheory.hasSum_lintegral_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {ι : Type.{u2}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), HasSum.{0, u2} ENNReal ι (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (μ i) (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.sum.{u1, u2} α ι m μ) (fun (a : α) => f a)) +but is expected to have type + forall {α : Type.{u1}} {ι : Type.{u2}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), HasSum.{0, u2} ENNReal ι (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (μ i) (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.sum.{u1, u2} α ι m μ) (fun (a : α) => f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.has_sum_lintegral_measure MeasureTheory.hasSum_lintegral_measureₓ'. -/ theorem hasSum_lintegral_measure {ι} {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) : HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) := (lintegral_sum_measure f μ).symm ▸ ENNReal.summable.HasSum #align measure_theory.has_sum_lintegral_measure MeasureTheory.hasSum_lintegral_measure +/- warning: measure_theory.lintegral_add_measure -> MeasureTheory.lintegral_add_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : MeasureTheory.Measure.{u1} α m) (ν : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHAdd.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instAdd.{u1} α m)) μ ν) (fun (a : α) => f a)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m ν (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal) (μ : MeasureTheory.Measure.{u1} α m) (ν : MeasureTheory.Measure.{u1} α m), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (HAdd.hAdd.{u1, u1, u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHAdd.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instAdd.{u1} α m)) μ ν) (fun (a : α) => f a)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m ν (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_measure MeasureTheory.lintegral_add_measureₓ'. -/ @[simp] theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) : (∫⁻ a, f a ∂μ + ν) = (∫⁻ a, f a ∂μ) + ∫⁻ a, f a ∂ν := by simpa [tsum_fintype] using lintegral_sum_measure f fun b => cond b μ ν #align measure_theory.lintegral_add_measure MeasureTheory.lintegral_add_measure +/- warning: measure_theory.lintegral_finset_sum_measure -> MeasureTheory.lintegral_finset_sum_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {ι : Type.{u2}} {m : MeasurableSpace.{u1} α} (s : Finset.{u2} ι) (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (Finset.sum.{u1, u2} (MeasureTheory.Measure.{u1} α m) ι (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) s (fun (i : ι) => μ i)) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal ι (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (μ i) (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {ι : Type.{u2}} {m : MeasurableSpace.{u1} α} (s : Finset.{u2} ι) (f : α -> ENNReal) (μ : ι -> (MeasureTheory.Measure.{u1} α m)), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (Finset.sum.{u1, u2} (MeasureTheory.Measure.{u1} α m) ι (MeasureTheory.Measure.instAddCommMonoid.{u1} α m) s (fun (i : ι) => μ i)) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal ι (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (i : ι) => MeasureTheory.lintegral.{u1} α m (μ i) (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_finset_sum_measure MeasureTheory.lintegral_finset_sum_measureₓ'. -/ @[simp] theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞) (μ : ι → Measure α) : (∫⁻ a, f a ∂∑ i in s, μ i) = ∑ i in s, ∫⁻ a, f a ∂μ i := @@ -681,26 +959,52 @@ theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset rfl #align measure_theory.lintegral_finset_sum_measure MeasureTheory.lintegral_finset_sum_measure +/- warning: measure_theory.lintegral_zero_measure -> MeasureTheory.lintegral_zero_measure is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m)))) (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m))) (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_zero_measure MeasureTheory.lintegral_zero_measureₓ'. -/ @[simp] theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂(0 : Measure α)) = 0 := bot_unique <| by simp [lintegral] #align measure_theory.lintegral_zero_measure MeasureTheory.lintegral_zero_measure +/- warning: measure_theory.set_lintegral_empty -> MeasureTheory.set_lintegral_empty is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) (fun (x : α) => f x)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) (fun (x : α) => f x)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_empty MeasureTheory.set_lintegral_emptyₓ'. -/ theorem set_lintegral_empty (f : α → ℝ≥0∞) : (∫⁻ x in ∅, f x ∂μ) = 0 := by rw [measure.restrict_empty, lintegral_zero_measure] #align measure_theory.set_lintegral_empty MeasureTheory.set_lintegral_empty +#print MeasureTheory.set_lintegral_univ /- theorem set_lintegral_univ (f : α → ℝ≥0∞) : (∫⁻ x in univ, f x ∂μ) = ∫⁻ x, f x ∂μ := by rw [measure.restrict_univ] #align measure_theory.set_lintegral_univ MeasureTheory.set_lintegral_univ +-/ +/- warning: measure_theory.set_lintegral_measure_zero -> MeasureTheory.set_lintegral_measure_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Set.{u1} α) (f : α -> ENNReal), (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Set.{u1} α) (f : α -> ENNReal), (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_measure_zero MeasureTheory.set_lintegral_measure_zeroₓ'. -/ theorem set_lintegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) : (∫⁻ x in s, f x ∂μ) = 0 := by convert lintegral_zero_measure _ exact measure.restrict_eq_zero.2 hs' #align measure_theory.set_lintegral_measure_zero MeasureTheory.set_lintegral_measure_zero +/- warning: measure_theory.lintegral_finset_sum' -> MeasureTheory.lintegral_finset_sum' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Finset.{u2} β) {f : β -> α -> ENNReal}, (forall (b : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f b) μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => f b a))) (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Finset.{u2} β) {f : β -> α -> ENNReal}, (forall (b : β), (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f b) μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => f b a))) (Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_finset_sum' MeasureTheory.lintegral_finset_sum'ₓ'. -/ theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, AEMeasurable (f b) μ) : (∫⁻ a, ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ a, f b a ∂μ := by @@ -711,11 +1015,23 @@ theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞} rw [lintegral_add_left' hf.1, ih hf.2] #align measure_theory.lintegral_finset_sum' MeasureTheory.lintegral_finset_sum' +/- warning: measure_theory.lintegral_finset_sum -> MeasureTheory.lintegral_finset_sum is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Finset.{u2} β) {f : β -> α -> ENNReal}, (forall (b : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f b))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => f b a))) (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (s : Finset.{u2} β) {f : β -> α -> ENNReal}, (forall (b : β), (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f b))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => f b a))) (Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_finset_sum MeasureTheory.lintegral_finset_sumₓ'. -/ theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) : (∫⁻ a, ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ a, f b a ∂μ := lintegral_finset_sum' s fun b hb => (hf b hb).AEMeasurable #align measure_theory.lintegral_finset_sum MeasureTheory.lintegral_finset_sum +/- warning: measure_theory.lintegral_const_mul -> MeasureTheory.lintegral_const_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const_mul MeasureTheory.lintegral_const_mulₓ'. -/ @[simp] theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, r * f a ∂μ) = r * ∫⁻ a, f a ∂μ := @@ -740,6 +1056,12 @@ theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measu #align measure_theory.lintegral_const_mul MeasureTheory.lintegral_const_mul +/- warning: measure_theory.lintegral_const_mul'' -> MeasureTheory.lintegral_const_mul'' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const_mul'' MeasureTheory.lintegral_const_mul''ₓ'. -/ theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : (∫⁻ a, r * f a ∂μ) = r * ∫⁻ a, f a ∂μ := by @@ -749,6 +1071,12 @@ theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEM rw [A, B, lintegral_const_mul _ hf.measurable_mk] #align measure_theory.lintegral_const_mul'' MeasureTheory.lintegral_const_mul'' +/- warning: measure_theory.lintegral_const_mul_le -> MeasureTheory.lintegral_const_mul_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const_mul_le MeasureTheory.lintegral_const_mul_leₓ'. -/ theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) : (r * ∫⁻ a, f a ∂μ) ≤ ∫⁻ a, r * f a ∂μ := by rw [lintegral, ENNReal.mul_iSup] @@ -761,6 +1089,12 @@ theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) : (r * exact mul_le_mul_left' (hs x) _ #align measure_theory.lintegral_const_mul_le MeasureTheory.lintegral_const_mul_le +/- warning: measure_theory.lintegral_const_mul' -> MeasureTheory.lintegral_const_mul' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_const_mul' MeasureTheory.lintegral_const_mul'ₓ'. -/ theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : (∫⁻ a, r * f a ∂μ) = r * ∫⁻ a, f a ∂μ := by @@ -776,22 +1110,52 @@ theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r #align measure_theory.lintegral_const_mul' MeasureTheory.lintegral_const_mul' +/- warning: measure_theory.lintegral_mul_const -> MeasureTheory.lintegral_mul_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mul_const MeasureTheory.lintegral_mul_constₓ'. -/ theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f a * r ∂μ) = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul r hf] #align measure_theory.lintegral_mul_const MeasureTheory.lintegral_mul_const +/- warning: measure_theory.lintegral_mul_const'' -> MeasureTheory.lintegral_mul_const'' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mul_const'' MeasureTheory.lintegral_mul_const''ₓ'. -/ theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : (∫⁻ a, f a * r ∂μ) = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul'' r hf] #align measure_theory.lintegral_mul_const'' MeasureTheory.lintegral_mul_const'' +/- warning: measure_theory.lintegral_mul_const_le -> MeasureTheory.lintegral_mul_const_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) r)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) r)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mul_const_le MeasureTheory.lintegral_mul_const_leₓ'. -/ theorem lintegral_mul_const_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂μ) * r ≤ ∫⁻ a, f a * r ∂μ := by simp_rw [mul_comm, lintegral_const_mul_le r f] #align measure_theory.lintegral_mul_const_le MeasureTheory.lintegral_mul_const_le +/- warning: measure_theory.lintegral_mul_const' -> MeasureTheory.lintegral_mul_const' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) r)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) r)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_mul_const' MeasureTheory.lintegral_mul_const'ₓ'. -/ theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : (∫⁻ a, f a * r ∂μ) = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul' r f hr] #align measure_theory.lintegral_mul_const' MeasureTheory.lintegral_mul_const' +/- warning: measure_theory.lintegral_lintegral_mul -> MeasureTheory.lintegral_lintegral_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u2} β] {ν : MeasureTheory.Measure.{u2} β _inst_1} {f : α -> ENNReal} {g : β -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (AEMeasurable.{u2, 0} β ENNReal ENNReal.measurableSpace _inst_1 g ν) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => MeasureTheory.lintegral.{u2} β _inst_1 ν (fun (y : β) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f x) (g y)))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u2} β _inst_1 ν (fun (y : β) => g y)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u2} β] {ν : MeasureTheory.Measure.{u2} β _inst_1} {f : α -> ENNReal} {g : β -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (AEMeasurable.{u2, 0} β ENNReal ENNReal.measurableSpace _inst_1 g ν) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => MeasureTheory.lintegral.{u2} β _inst_1 ν (fun (y : β) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f x) (g y)))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u2} β _inst_1 ν (fun (y : β) => g y)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_lintegral_mul MeasureTheory.lintegral_lintegral_mulₓ'. -/ /- A double integral of a product where each factor contains only one variable is a product of integrals -/ theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ≥0∞} @@ -800,18 +1164,36 @@ theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : simp [lintegral_const_mul'' _ hg, lintegral_mul_const'' _ hf] #align measure_theory.lintegral_lintegral_mul MeasureTheory.lintegral_lintegral_mul +/- warning: measure_theory.lintegral_rw₁ -> MeasureTheory.lintegral_rw₁ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> β} {f' : α -> β}, (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f f') -> (forall (g : β -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g (f a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g (f' a)))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} {m : MeasurableSpace.{u2} α} {μ : MeasureTheory.Measure.{u2} α m} {f : α -> β} {f' : α -> β}, (Filter.EventuallyEq.{u2, u1} α β (MeasureTheory.Measure.ae.{u2} α m μ) f f') -> (forall (g : β -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u2} α m μ (fun (a : α) => g (f a))) (MeasureTheory.lintegral.{u2} α m μ (fun (a : α) => g (f' a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_rw₁ MeasureTheory.lintegral_rw₁ₓ'. -/ -- TODO: Need a better way of rewriting inside of a integral theorem lintegral_rw₁ {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞) : (∫⁻ a, g (f a) ∂μ) = ∫⁻ a, g (f' a) ∂μ := lintegral_congr_ae <| h.mono fun a h => by rw [h] #align measure_theory.lintegral_rw₁ MeasureTheory.lintegral_rw₁ +/- warning: measure_theory.lintegral_rw₂ -> MeasureTheory.lintegral_rw₂ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f₁ : α -> β} {f₁' : α -> β} {f₂ : α -> γ} {f₂' : α -> γ}, (Filter.EventuallyEq.{u1, u2} α β (MeasureTheory.Measure.ae.{u1} α m μ) f₁ f₁') -> (Filter.EventuallyEq.{u1, u3} α γ (MeasureTheory.Measure.ae.{u1} α m μ) f₂ f₂') -> (forall (g : β -> γ -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g (f₁ a) (f₂ a))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g (f₁' a) (f₂' a)))) +but is expected to have type + forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} {m : MeasurableSpace.{u3} α} {μ : MeasureTheory.Measure.{u3} α m} {f₁ : α -> β} {f₁' : α -> β} {f₂ : α -> γ} {f₂' : α -> γ}, (Filter.EventuallyEq.{u3, u2} α β (MeasureTheory.Measure.ae.{u3} α m μ) f₁ f₁') -> (Filter.EventuallyEq.{u3, u1} α γ (MeasureTheory.Measure.ae.{u3} α m μ) f₂ f₂') -> (forall (g : β -> γ -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u3} α m μ (fun (a : α) => g (f₁ a) (f₂ a))) (MeasureTheory.lintegral.{u3} α m μ (fun (a : α) => g (f₁' a) (f₂' a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_rw₂ MeasureTheory.lintegral_rw₂ₓ'. -/ -- TODO: Need a better way of rewriting inside of a integral theorem lintegral_rw₂ {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂') (g : β → γ → ℝ≥0∞) : (∫⁻ a, g (f₁ a) (f₂ a) ∂μ) = ∫⁻ a, g (f₁' a) (f₂' a) ∂μ := lintegral_congr_ae <| h₁.mp <| h₂.mono fun _ h₂ h₁ => by rw [h₁, h₂] #align measure_theory.lintegral_rw₂ MeasureTheory.lintegral_rw₂ +/- warning: measure_theory.lintegral_indicator -> MeasureTheory.lintegral_indicator is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal ENNReal.hasZero s f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal instENNRealZero s f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_indicator MeasureTheory.lintegral_indicatorₓ'. -/ @[simp] theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : (∫⁻ a, s.indicator f a ∂μ) = ∫⁻ a in s, f a ∂μ := @@ -828,6 +1210,12 @@ theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : Measurabl simp [hφ x, hs, indicator_le_indicator] #align measure_theory.lintegral_indicator MeasureTheory.lintegral_indicator +/- warning: measure_theory.lintegral_indicator₀ -> MeasureTheory.lintegral_indicator₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {s : Set.{u1} α}, (MeasureTheory.NullMeasurableSet.{u1} α m s μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal ENNReal.hasZero s f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {s : Set.{u1} α}, (MeasureTheory.NullMeasurableSet.{u1} α m s μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal instENNRealZero s f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_indicator₀ MeasureTheory.lintegral_indicator₀ₓ'. -/ theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) : (∫⁻ a, s.indicator f a ∂μ) = ∫⁻ a in s, f a ∂μ := by rw [← lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.to_measurable_ae_eq), @@ -835,11 +1223,23 @@ theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMe measure.restrict_congr_set hs.to_measurable_ae_eq] #align measure_theory.lintegral_indicator₀ MeasureTheory.lintegral_indicator₀ +/- warning: measure_theory.lintegral_indicator_const -> MeasureTheory.lintegral_indicator_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (forall (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal ENNReal.hasZero s (fun (_x : α) => c) a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (forall (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u1, 0} α ENNReal instENNRealZero s (fun (_x : α) => c) a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_indicator_const MeasureTheory.lintegral_indicator_constₓ'. -/ theorem lintegral_indicator_const {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞) : (∫⁻ a, s.indicator (fun _ => c) a ∂μ) = c * μ s := by rw [lintegral_indicator _ hs, set_lintegral_const] #align measure_theory.lintegral_indicator_const MeasureTheory.lintegral_indicator_const +/- warning: measure_theory.set_lintegral_eq_const -> MeasureTheory.set_lintegral_eq_const is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (r : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) r))) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) r))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (r : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) r))) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) r))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_eq_const MeasureTheory.set_lintegral_eq_constₓ'. -/ theorem set_lintegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) : (∫⁻ x in { x | f x = r }, f x ∂μ) = r * μ { x | f x = r } := by @@ -850,6 +1250,12 @@ theorem set_lintegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : exact hf (measurable_set_singleton r) #align measure_theory.set_lintegral_eq_const MeasureTheory.set_lintegral_eq_const +/- warning: measure_theory.lintegral_add_mul_meas_add_le_le_lintegral -> MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) ε (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f x) ε) (g x)))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) ε (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (f x) ε) (g x)))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_mul_meas_add_le_le_lintegral MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegralₓ'. -/ /-- A version of **Markov's inequality** for two functions. It doesn't follow from the standard Markov's inequality because we only assume measurability of `g`, not `f`. -/ theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g) @@ -873,6 +1279,12 @@ theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hl exacts[hx₂, (add_zero _).trans_le <| (hφ_le x).trans hx₁] #align measure_theory.lintegral_add_mul_meas_add_le_le_lintegral MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral +/- warning: measure_theory.mul_meas_ge_le_lintegral₀ -> MeasureTheory.mul_meas_ge_le_lintegral₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) ε (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (f x))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) ε (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (f x))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.mul_meas_ge_le_lintegral₀ MeasureTheory.mul_meas_ge_le_lintegral₀ₓ'. -/ /-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/ theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) : ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ := by @@ -880,6 +1292,12 @@ theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf ε #align measure_theory.mul_meas_ge_le_lintegral₀ MeasureTheory.mul_meas_ge_le_lintegral₀ +/- warning: measure_theory.mul_meas_ge_le_lintegral -> MeasureTheory.mul_meas_ge_le_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) ε (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (f x))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (ε : ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) ε (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (f x))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.mul_meas_ge_le_lintegral MeasureTheory.mul_meas_ge_le_lintegralₓ'. -/ /-- **Markov's inequality** also known as **Chebyshev's first inequality**. For a version assuming `ae_measurable`, see `mul_meas_ge_le_lintegral₀`. -/ theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞) : @@ -887,6 +1305,12 @@ theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε mul_meas_ge_le_lintegral₀ hf.AEMeasurable ε #align measure_theory.mul_meas_ge_le_lintegral MeasureTheory.mul_meas_ge_le_lintegral +/- warning: measure_theory.lintegral_eq_top_of_measure_eq_top_pos -> MeasureTheory.lintegral_eq_top_of_measure_eq_top_pos is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => Eq.{1} ENNReal (f x) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_eq_top_of_measure_eq_top_pos MeasureTheory.lintegral_eq_top_of_measure_eq_top_posₓ'. -/ theorem lintegral_eq_top_of_measure_eq_top_pos {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hμf : 0 < μ { x | f x = ∞ }) : (∫⁻ x, f x ∂μ) = ∞ := eq_top_iff.mpr <| @@ -896,6 +1320,12 @@ theorem lintegral_eq_top_of_measure_eq_top_pos {f : α → ℝ≥0∞} (hf : AEM #align measure_theory.lintegral_eq_top_of_measure_eq_top_pos MeasureTheory.lintegral_eq_top_of_measure_eq_top_pos +/- warning: measure_theory.meas_ge_le_lintegral_div -> MeasureTheory.meas_ge_le_lintegral_div is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Ne.{1} ENNReal ε (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (f x)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toHasDiv.{0} ENNReal ENNReal.divInvMonoid)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) ε))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Ne.{1} ENNReal ε (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (f x)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toDiv.{0} ENNReal ENNReal.instDivInvMonoidENNReal)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) ε))) +Case conversion may be inaccurate. Consider using '#align measure_theory.meas_ge_le_lintegral_div MeasureTheory.meas_ge_le_lintegral_divₓ'. -/ /-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/ theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) (hε' : ε ≠ ∞) : μ { x | ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε := @@ -905,6 +1335,12 @@ theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ exact mul_meas_ge_le_lintegral₀ hf ε #align measure_theory.meas_ge_le_lintegral_div MeasureTheory.meas_ge_le_lintegral_div +/- warning: measure_theory.ae_eq_of_ae_le_of_lintegral_le -> MeasureTheory.ae_eq_of_ae_le_of_lintegral_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) -> (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f g) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) -> (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f g) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_eq_of_ae_le_of_lintegral_le MeasureTheory.ae_eq_of_ae_le_of_lintegral_leₓ'. -/ theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : (∫⁻ x, f x ∂μ) ≠ ∞) (hg : AEMeasurable g μ) (hgf : (∫⁻ x, g x ∂μ) ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g := by @@ -922,6 +1358,12 @@ theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top) #align measure_theory.ae_eq_of_ae_le_of_lintegral_le MeasureTheory.ae_eq_of_ae_le_of_lintegral_le +/- warning: measure_theory.lintegral_eq_zero_iff' -> MeasureTheory.lintegral_eq_zero_iff' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Iff (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (OfNat.mk.{u1} (α -> ENNReal) 0 (Zero.zero.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => ENNReal.hasZero))))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Iff (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (Zero.toOfNat0.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => ENNReal) (fun (i : α) => instENNRealZero)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_eq_zero_iff' MeasureTheory.lintegral_eq_zero_iff'ₓ'. -/ @[simp] theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : (∫⁻ a, f a ∂μ) = 0 ↔ f =ᵐ[μ] 0 := @@ -932,16 +1374,34 @@ theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) fun h => (lintegral_congr_ae h).trans lintegral_zero⟩ #align measure_theory.lintegral_eq_zero_iff' MeasureTheory.lintegral_eq_zero_iff' +/- warning: measure_theory.lintegral_eq_zero_iff -> MeasureTheory.lintegral_eq_zero_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (OfNat.mk.{u1} (α -> ENNReal) 0 (Zero.zero.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => ENNReal.hasZero))))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (Zero.toOfNat0.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => ENNReal) (fun (i : α) => instENNRealZero)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_eq_zero_iff MeasureTheory.lintegral_eq_zero_iffₓ'. -/ @[simp] theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f a ∂μ) = 0 ↔ f =ᵐ[μ] 0 := lintegral_eq_zero_iff' hf.AEMeasurable #align measure_theory.lintegral_eq_zero_iff MeasureTheory.lintegral_eq_zero_iff +/- warning: measure_theory.lintegral_pos_iff_support -> MeasureTheory.lintegral_pos_iff_support is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Function.support.{u1, 0} α ENNReal ENNReal.hasZero f)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a))) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Function.support.{u1, 0} α ENNReal instENNRealZero f)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_pos_iff_support MeasureTheory.lintegral_pos_iff_supportₓ'. -/ theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) : (0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) := by simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support] #align measure_theory.lintegral_pos_iff_support MeasureTheory.lintegral_pos_iff_support +/- warning: measure_theory.lintegral_supr_ae -> MeasureTheory.lintegral_iSup_ae is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f n a) (f (Nat.succ n) a)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f n a) (f (Nat.succ n) a)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n a))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_supr_ae MeasureTheory.lintegral_iSup_aeₓ'. -/ /-- Weaker version of the monotone convergence theorem-/ theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : (∫⁻ a, ⨆ n, f n a ∂μ) = ⨆ n, ∫⁻ a, f n a ∂μ := @@ -963,6 +1423,12 @@ theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurabl #align measure_theory.lintegral_supr_ae MeasureTheory.lintegral_iSup_ae +/- warning: measure_theory.lintegral_sub' -> MeasureTheory.lintegral_sub' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) g f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (f a) (g a))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) g f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (f a) (g a))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_sub' MeasureTheory.lintegral_sub'ₓ'. -/ theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : (∫⁻ a, g a ∂μ) ≠ ∞) (h_le : g ≤ᵐ[μ] f) : (∫⁻ a, f a - g a ∂μ) = (∫⁻ a, f a ∂μ) - ∫⁻ a, g a ∂μ := by @@ -971,11 +1437,23 @@ theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fi exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx) #align measure_theory.lintegral_sub' MeasureTheory.lintegral_sub' +/- warning: measure_theory.lintegral_sub -> MeasureTheory.lintegral_sub is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) g f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (f a) (g a))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) g f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (f a) (g a))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_sub MeasureTheory.lintegral_subₓ'. -/ theorem lintegral_sub {f g : α → ℝ≥0∞} (hg : Measurable g) (hg_fin : (∫⁻ a, g a ∂μ) ≠ ∞) (h_le : g ≤ᵐ[μ] f) : (∫⁻ a, f a - g a ∂μ) = (∫⁻ a, f a ∂μ) - ∫⁻ a, g a ∂μ := lintegral_sub' hg.AEMeasurable hg_fin h_le #align measure_theory.lintegral_sub MeasureTheory.lintegral_sub +/- warning: measure_theory.lintegral_sub_le' -> MeasureTheory.lintegral_sub_le' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (g x) (f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (g x) (f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_sub_le' MeasureTheory.lintegral_sub_le'ₓ'. -/ theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) : ((∫⁻ x, g x ∂μ) - ∫⁻ x, f x ∂μ) ≤ ∫⁻ x, g x - f x ∂μ := by @@ -987,11 +1465,23 @@ theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) : exact lintegral_mono fun x => le_tsub_add #align measure_theory.lintegral_sub_le' MeasureTheory.lintegral_sub_le' +/- warning: measure_theory.lintegral_sub_le -> MeasureTheory.lintegral_sub_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.hasSub) (g x) (f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) (g : α -> ENNReal), (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => HSub.hSub.{0, 0, 0} ENNReal ENNReal ENNReal (instHSub.{0} ENNReal ENNReal.instSub) (g x) (f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_sub_le MeasureTheory.lintegral_sub_leₓ'. -/ theorem lintegral_sub_le (f g : α → ℝ≥0∞) (hf : Measurable f) : ((∫⁻ x, g x ∂μ) - ∫⁻ x, f x ∂μ) ≤ ∫⁻ x, g x - f x ∂μ := lintegral_sub_le' f g hf.AEMeasurable #align measure_theory.lintegral_sub_le MeasureTheory.lintegral_sub_le +/- warning: measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt -> MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Filter.Frequently.{u1} α (fun (x : α) => Ne.{1} ENNReal (f x) (g x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (Filter.Frequently.{u1} α (fun (x : α) => Ne.{1} ENNReal (f x) (g x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_ltₓ'. -/ theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) : (∫⁻ x, f x ∂μ) < ∫⁻ x, g x ∂μ := by @@ -1000,6 +1490,12 @@ theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0 exact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h #align measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt +/- warning: measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on -> MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (forall {s : Set.{u1} α}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) f g) -> (forall {s : Set.{u1} α}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_onₓ'. -/ theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0) (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : (∫⁻ x, f x ∂μ) < ∫⁻ x, g x ∂μ := @@ -1007,6 +1503,12 @@ theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg ((frequently_ae_mem_iff.2 hμs).and_eventually h).mono fun x hx => (hx.2 hx.1).Ne #align measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on +/- warning: measure_theory.lintegral_strict_mono -> MeasureTheory.lintegral_strict_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Ne.{succ u1} (MeasureTheory.Measure.{u1} α m) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m))))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Ne.{succ u1} (MeasureTheory.Measure.{u1} α m) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m)))) -> (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x)) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => g x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_strict_mono MeasureTheory.lintegral_strict_monoₓ'. -/ theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : (∫⁻ x, f x ∂μ) < ∫⁻ x, g x ∂μ := by @@ -1015,12 +1517,24 @@ theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : A simpa using h #align measure_theory.lintegral_strict_mono MeasureTheory.lintegral_strict_mono +/- warning: measure_theory.set_lintegral_strict_mono -> MeasureTheory.set_lintegral_strict_mono is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal} {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal} {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (x : α) => (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => g x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_strict_mono MeasureTheory.set_lintegral_strict_monoₓ'. -/ theorem set_lintegral_strict_mono {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s) (hs : μ s ≠ 0) (hg : Measurable g) (hfi : (∫⁻ x in s, f x ∂μ) ≠ ∞) (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : (∫⁻ x in s, f x ∂μ) < ∫⁻ x in s, g x ∂μ := lintegral_strict_mono (by simp [hs]) hg.AEMeasurable hfi ((ae_restrict_iff' hsm).mpr h) #align measure_theory.set_lintegral_strict_mono MeasureTheory.set_lintegral_strict_mono +/- warning: measure_theory.lintegral_infi_ae -> MeasureTheory.lintegral_iInf_ae is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) (f (Nat.succ n)) (f n)) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n a))) (iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) (f (Nat.succ n)) (f n)) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n a))) (iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_infi_ae MeasureTheory.lintegral_iInf_aeₓ'. -/ /-- Monotone convergence theorem for nonincreasing sequences of functions -/ theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_mono : ∀ n : ℕ, f n.succ ≤ᵐ[μ] f n) (h_fin : (∫⁻ a, f 0 a ∂μ) ≠ ∞) : @@ -1053,12 +1567,24 @@ theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measu #align measure_theory.lintegral_infi_ae MeasureTheory.lintegral_iInf_ae +/- warning: measure_theory.lintegral_infi -> MeasureTheory.lintegral_iInf is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (Antitone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) f) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f n a))) (iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (Antitone.{0, u1} Nat (α -> ENNReal) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (Pi.preorder.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) f) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f n a))) (iInf.{0, 1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_infi MeasureTheory.lintegral_iInfₓ'. -/ /-- Monotone convergence theorem for nonincreasing sequences of functions -/ theorem lintegral_iInf {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_anti : Antitone f) (h_fin : (∫⁻ a, f 0 a ∂μ) ≠ ∞) : (∫⁻ a, ⨅ n, f n a ∂μ) = ⨅ n, ∫⁻ a, f n a ∂μ := lintegral_iInf_ae h_meas (fun n => ae_of_all _ <| h_anti n.le_succ) h_fin #align measure_theory.lintegral_infi MeasureTheory.lintegral_iInf +/- warning: measure_theory.lintegral_liminf_le' -> MeasureTheory.lintegral_liminf_le' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.liminf.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))))) (Filter.liminf.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f n) μ) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.liminf.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))))) (Filter.liminf.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_liminf_le' MeasureTheory.lintegral_liminf_le'ₓ'. -/ /-- Known as Fatou's lemma, version with `ae_measurable` functions -/ theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) : (∫⁻ a, liminf (fun n => f n a) atTop ∂μ) ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop := @@ -1066,26 +1592,38 @@ theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AE (∫⁻ a, liminf (fun n => f n a) atTop ∂μ) = ∫⁻ a, ⨆ n : ℕ, ⨅ i ≥ n, f i a ∂μ := by simp only [liminf_eq_supr_infi_of_nat] _ = ⨆ n : ℕ, ∫⁻ a, ⨅ i ≥ n, f i a ∂μ := - (lintegral_supr' (fun n => aemeasurable_biInf _ (to_countable _) h_meas) + (lintegral_iSup' (fun n => aemeasurable_biInf _ (to_countable _) h_meas) (ae_of_all μ fun a n m hnm => iInf_le_iInf_of_subset fun i hi => le_trans hnm hi)) - _ ≤ ⨆ n : ℕ, ⨅ i ≥ n, ∫⁻ a, f i a ∂μ := (iSup_mono fun n => le_infi₂_lintegral _) + _ ≤ ⨆ n : ℕ, ⨅ i ≥ n, ∫⁻ a, f i a ∂μ := (iSup_mono fun n => le_iInf₂_lintegral _) _ = atTop.liminf fun n => ∫⁻ a, f n a ∂μ := Filter.liminf_eq_iSup_iInf_of_nat.symm #align measure_theory.lintegral_liminf_le' MeasureTheory.lintegral_liminf_le' +/- warning: measure_theory.lintegral_liminf_le -> MeasureTheory.lintegral_liminf_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.liminf.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))))) (Filter.liminf.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.liminf.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))))) (Filter.liminf.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_liminf_le MeasureTheory.lintegral_liminf_leₓ'. -/ /-- Known as Fatou's lemma -/ theorem lintegral_liminf_le {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) : (∫⁻ a, liminf (fun n => f n a) atTop ∂μ) ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop := lintegral_liminf_le' fun n => (h_meas n).AEMeasurable #align measure_theory.lintegral_liminf_le MeasureTheory.lintegral_liminf_le +/- warning: measure_theory.limsup_lintegral_le -> MeasureTheory.limsup_lintegral_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal} {g : α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) (f n) g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (Filter.limsup.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.limsup.{0, 0} ENNReal Nat (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : Nat -> α -> ENNReal} {g : α -> ENNReal}, (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) (f n) g) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => g a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (Filter.limsup.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Filter.limsup.{0, 0} ENNReal Nat (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))) (fun (n : Nat) => f n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.limsup_lintegral_le MeasureTheory.limsup_lintegral_leₓ'. -/ theorem limsup_lintegral_le {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} (hf_meas : ∀ n, Measurable (f n)) (h_bound : ∀ n, f n ≤ᵐ[μ] g) (h_fin : (∫⁻ a, g a ∂μ) ≠ ∞) : limsup (fun n => ∫⁻ a, f n a ∂μ) atTop ≤ ∫⁻ a, limsup (fun n => f n a) atTop ∂μ := calc limsup (fun n => ∫⁻ a, f n a ∂μ) atTop = ⨅ n : ℕ, ⨆ i ≥ n, ∫⁻ a, f i a ∂μ := limsup_eq_iInf_iSup_of_nat - _ ≤ ⨅ n : ℕ, ∫⁻ a, ⨆ i ≥ n, f i a ∂μ := (iInf_mono fun n => supr₂_lintegral_le _) + _ ≤ ⨅ n : ℕ, ∫⁻ a, ⨆ i ≥ n, f i a ∂μ := (iInf_mono fun n => iSup₂_lintegral_le _) _ = ∫⁻ a, ⨅ n : ℕ, ⨆ i ≥ n, f i a ∂μ := by refine' (lintegral_infi _ _ _).symm @@ -1100,6 +1638,12 @@ theorem limsup_lintegral_le {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0 #align measure_theory.limsup_lintegral_le MeasureTheory.limsup_lintegral_le +/- warning: measure_theory.tendsto_lintegral_of_dominated_convergence -> MeasureTheory.tendsto_lintegral_of_dominated_convergence is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {F : Nat -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (F n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) (F n) bound) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => F n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {F : Nat -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (forall (n : Nat), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (F n)) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) (F n) bound) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => F n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_lintegral_of_dominated_convergence MeasureTheory.tendsto_lintegral_of_dominated_convergenceₓ'. -/ /-- Dominated convergence theorem for nonnegative functions -/ theorem tendsto_lintegral_of_dominated_convergence {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ n, Measurable (F n)) (h_bound : ∀ n, F n ≤ᵐ[μ] bound) @@ -1118,6 +1662,12 @@ theorem tendsto_lintegral_of_dominated_convergence {F : ℕ → α → ℝ≥0 ) #align measure_theory.tendsto_lintegral_of_dominated_convergence MeasureTheory.tendsto_lintegral_of_dominated_convergence +/- warning: measure_theory.tendsto_lintegral_of_dominated_convergence' -> MeasureTheory.tendsto_lintegral_of_dominated_convergence' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {F : Nat -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (F n) μ) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.Measure.ae.{u1} α m μ) (F n) bound) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => F n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{0} ENNReal ENNReal.topologicalSpace (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {F : Nat -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (forall (n : Nat), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (F n) μ) -> (forall (n : Nat), Filter.EventuallyLE.{u1, 0} α ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.Measure.ae.{u1} α m μ) (F n) bound) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => F n a) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{0, 0} Nat ENNReal (fun (n : Nat) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_lintegral_of_dominated_convergence' MeasureTheory.tendsto_lintegral_of_dominated_convergence'ₓ'. -/ /-- Dominated convergence theorem for nonnegative functions which are just almost everywhere measurable. -/ theorem tendsto_lintegral_of_dominated_convergence' {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} @@ -1140,6 +1690,12 @@ theorem tendsto_lintegral_of_dominated_convergence' {F : ℕ → α → ℝ≥0 rwa [H'] at H #align measure_theory.tendsto_lintegral_of_dominated_convergence' MeasureTheory.tendsto_lintegral_of_dominated_convergence' +/- warning: measure_theory.tendsto_lintegral_filter_of_dominated_convergence -> MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Type.{u2}} {l : Filter.{u2} ι} [_inst_1 : Filter.IsCountablyGenerated.{u2} ι l] {F : ι -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (Filter.Eventually.{u2} ι (fun (n : ι) => Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (F n)) l) -> (Filter.Eventually.{u2} ι (fun (n : ι) => Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (F n a) (bound a)) (MeasureTheory.Measure.ae.{u1} α m μ)) l) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{u2, 0} ι ENNReal (fun (n : ι) => F n a) l (nhds.{0} ENNReal ENNReal.topologicalSpace (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{u2, 0} ι ENNReal (fun (n : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) l (nhds.{0} ENNReal ENNReal.topologicalSpace (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {ι : Type.{u2}} {l : Filter.{u2} ι} [_inst_1 : Filter.IsCountablyGenerated.{u2} ι l] {F : ι -> α -> ENNReal} {f : α -> ENNReal} (bound : α -> ENNReal), (Filter.Eventually.{u2} ι (fun (n : ι) => Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (F n)) l) -> (Filter.Eventually.{u2} ι (fun (n : ι) => Filter.Eventually.{u1} α (fun (a : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (F n a) (bound a)) (MeasureTheory.Measure.ae.{u1} α m μ)) l) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => bound a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (a : α) => Filter.Tendsto.{u2, 0} ι ENNReal (fun (n : ι) => F n a) l (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (f a))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (Filter.Tendsto.{u2, 0} ι ENNReal (fun (n : ι) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => F n a)) l (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_lintegral_filter_of_dominated_convergence MeasureTheory.tendsto_lintegral_filter_of_dominated_convergenceₓ'. -/ /-- Dominated convergence theorem for filters with a countable basis -/ theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) @@ -1176,6 +1732,12 @@ section open Encodable +/- warning: measure_theory.lintegral_supr_directed_of_measurable -> MeasureTheory.lintegral_iSup_directed_of_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (b : β), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f b)) -> (Directed.{u1, succ u2} (α -> ENNReal) β (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) β (fun (b : β) => f b a))) (iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) β (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (b : β), Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace (f b)) -> (Directed.{u1, succ u2} (α -> ENNReal) β (fun (x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25151 : α -> ENNReal) (x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25153 : α -> ENNReal) => LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (a._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25134 : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25151 x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25153) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) β (fun (b : β) => f b a))) (iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) β (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_supr_directed_of_measurable MeasureTheory.lintegral_iSup_directed_of_measurableₓ'. -/ /-- Monotone convergence for a supremum over a directed family and indexed by a countable type -/ theorem lintegral_iSup_directed_of_measurable [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, Measurable (f b)) (h_directed : Directed (· ≤ ·) f) : @@ -1202,6 +1764,12 @@ theorem lintegral_iSup_directed_of_measurable [Countable β] {f : β → α → #align measure_theory.lintegral_supr_directed_of_measurable MeasureTheory.lintegral_iSup_directed_of_measurable +/- warning: measure_theory.lintegral_supr_directed -> MeasureTheory.lintegral_iSup_directed is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (b : β), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f b) μ) -> (Directed.{u1, succ u2} (α -> ENNReal) β (LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))))) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) β (fun (b : β) => f b a))) (iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) β (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (b : β), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f b) μ) -> (Directed.{u1, succ u2} (α -> ENNReal) β (fun (x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25659 : α -> ENNReal) (x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25661 : α -> ENNReal) => LE.le.{u1} (α -> ENNReal) (Pi.hasLe.{u1, 0} α (fun (a._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25641 : α) => ENNReal) (fun (i : α) => Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25659 x._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.25661) f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) β (fun (b : β) => f b a))) (iSup.{0, succ u2} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) β (fun (b : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f b a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_supr_directed MeasureTheory.lintegral_iSup_directedₓ'. -/ /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:38: in filter_upwards #[[], ["with", ident x, ident i, ident j], []]: ./././Mathport/Syntax/Translate/Basic.lean:349:22: unsupported: parse error @ arg 0: next failed, no more args -/ /-- Monotone convergence for a supremum over a directed family and indexed by a countable type. -/ theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ) @@ -1238,6 +1806,12 @@ theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (h end +/- warning: measure_theory.lintegral_tsum -> MeasureTheory.lintegral_tsum is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (i : β), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f i) μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace β (fun (i : β) => f i a))) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace β (fun (i : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {f : β -> α -> ENNReal}, (forall (i : β), AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (f i) μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal β (fun (i : β) => f i a))) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal β (fun (i : β) => MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f i a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_tsum MeasureTheory.lintegral_tsumₓ'. -/ theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) : (∫⁻ a, ∑' i, f i a ∂μ) = ∑' i, ∫⁻ a, f i a ∂μ := by @@ -1255,44 +1829,86 @@ theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i open Measure -theorem lintegral_Union₀ [Countable β] {s : β → Set α} (hm : ∀ i, NullMeasurableSet (s i) μ) +/- warning: measure_theory.lintegral_Union₀ -> MeasureTheory.lintegral_iUnion₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), MeasureTheory.NullMeasurableSet.{u1} α m (s i) μ) -> (Pairwise.{u2} β (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), MeasureTheory.NullMeasurableSet.{u1} α m (s i) μ) -> (Pairwise.{u2} β (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_Union₀ MeasureTheory.lintegral_iUnion₀ₓ'. -/ +theorem lintegral_iUnion₀ [Countable β] {s : β → Set α} (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ i, s i, f a ∂μ) = ∑' i, ∫⁻ a in s i, f a ∂μ := by simp only [measure.restrict_Union_ae hd hm, lintegral_sum_measure] -#align measure_theory.lintegral_Union₀ MeasureTheory.lintegral_Union₀ - +#align measure_theory.lintegral_Union₀ MeasureTheory.lintegral_iUnion₀ + +/- warning: measure_theory.lintegral_Union -> MeasureTheory.lintegral_iUnion is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), MeasurableSet.{u1} α m (s i)) -> (Pairwise.{u2} β (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)))) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), MeasurableSet.{u1} α m (s i)) -> (Pairwise.{u2} β (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (Disjoint.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α))))))) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_Union MeasureTheory.lintegral_iUnionₓ'. -/ theorem lintegral_iUnion [Countable β] {s : β → Set α} (hm : ∀ i, MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ i, s i, f a ∂μ) = ∑' i, ∫⁻ a in s i, f a ∂μ := - lintegral_Union₀ (fun i => (hm i).NullMeasurableSet) hd.AEDisjoint f + lintegral_iUnion₀ (fun i => (hm i).NullMeasurableSet) hd.AEDisjoint f #align measure_theory.lintegral_Union MeasureTheory.lintegral_iUnion -theorem lintegral_bUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable) +/- warning: measure_theory.lintegral_bUnion₀ -> MeasureTheory.lintegral_biUnion₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {t : Set.{u2} β} {s : β -> (Set.{u1} α)}, (Set.Countable.{u2} β t) -> (forall (i : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) -> (MeasureTheory.NullMeasurableSet.{u1} α m (s i) μ)) -> (Set.Pairwise.{u2} β t (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) => s i)))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) (fun (i : coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) β (HasLiftT.mk.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) β (CoeTCₓ.coe.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) β (coeBase.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) β (coeSubtype.{succ u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x t))))) i))) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {t : Set.{u2} β} {s : β -> (Set.{u1} α)}, (Set.Countable.{u2} β t) -> (forall (i : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) -> (MeasureTheory.NullMeasurableSet.{u1} α m (s i) μ)) -> (Set.Pairwise.{u2} β t (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => Set.iUnion.{u1, 0} α (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) (fun (H : Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) => s i)))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal (Set.Elem.{u2} β t) (fun (i : Set.Elem.{u2} β t) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s (Subtype.val.{succ u2} β (fun (x : β) => Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x t) i))) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_bUnion₀ MeasureTheory.lintegral_biUnion₀ₓ'. -/ +theorem lintegral_biUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, NullMeasurableSet (s i) μ) (hd : t.Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ) = ∑' i : t, ∫⁻ a in s i, f a ∂μ := by haveI := ht.to_encodable rw [bUnion_eq_Union, lintegral_Union₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)] -#align measure_theory.lintegral_bUnion₀ MeasureTheory.lintegral_bUnion₀ - -theorem lintegral_bUnion {t : Set β} {s : β → Set α} (ht : t.Countable) +#align measure_theory.lintegral_bUnion₀ MeasureTheory.lintegral_biUnion₀ + +/- warning: measure_theory.lintegral_bUnion -> MeasureTheory.lintegral_biUnion is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {t : Set.{u2} β} {s : β -> (Set.{u1} α)}, (Set.Countable.{u2} β t) -> (forall (i : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) -> (MeasurableSet.{u1} α m (s i))) -> (Set.PairwiseDisjoint.{u1, u2} (Set.{u1} α) β (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) t s) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) i t) => s i)))) (fun (a : α) => f a)) 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(coeBase.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) t) β (coeSubtype.{succ u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x t))))) i))) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {t : Set.{u2} β} {s : β -> (Set.{u1} α)}, (Set.Countable.{u2} β t) -> (forall (i : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) -> (MeasurableSet.{u1} α m (s i))) -> (Set.PairwiseDisjoint.{u1, u2} (Set.{u1} α) β (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) t s) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => Set.iUnion.{u1, 0} α (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) (fun (H : Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) i t) => s i)))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal (Set.Elem.{u2} β t) (fun (i : Set.Elem.{u2} β t) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s (Subtype.val.{succ u2} β (fun (x : β) => Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x t) i))) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_bUnion MeasureTheory.lintegral_biUnionₓ'. -/ +theorem lintegral_biUnion {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ) = ∑' i : t, ∫⁻ a in s i, f a ∂μ := - lintegral_bUnion₀ ht (fun i hi => (hm i hi).NullMeasurableSet) hd.AEDisjoint f -#align measure_theory.lintegral_bUnion MeasureTheory.lintegral_bUnion - -theorem lintegral_bUnion_finset₀ {s : Finset β} {t : β → Set α} + lintegral_biUnion₀ ht (fun i hi => (hm i hi).NullMeasurableSet) hd.AEDisjoint f +#align measure_theory.lintegral_bUnion MeasureTheory.lintegral_biUnion + +/- warning: measure_theory.lintegral_bUnion_finset₀ -> MeasureTheory.lintegral_biUnion_finset₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Finset.{u2} β} {t : β -> (Set.{u1} α)}, (Set.Pairwise.{u2} β ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) s) (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) t)) -> (forall (b : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) -> (MeasureTheory.NullMeasurableSet.{u1} α m (t b) μ)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (b : β) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) (fun (H : Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) => t b)))) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (t b)) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Finset.{u2} β} {t : β -> (Set.{u1} α)}, (Set.Pairwise.{u2} β (Finset.toSet.{u2} β s) (Function.onFun.{succ u2, succ u1, 1} β (Set.{u1} α) Prop (MeasureTheory.AEDisjoint.{u1} α m μ) t)) -> (forall (b : β), (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) -> (MeasureTheory.NullMeasurableSet.{u1} α m (t b) μ)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (b : β) => Set.iUnion.{u1, 0} α (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) (fun (H : Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) => t b)))) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (t b)) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_bUnion_finset₀ MeasureTheory.lintegral_biUnion_finset₀ₓ'. -/ +theorem lintegral_biUnion_finset₀ {s : Finset β} {t : β → Set α} (hd : Set.Pairwise (↑s) (AEDisjoint μ on t)) (hm : ∀ b ∈ s, NullMeasurableSet (t b) μ) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ) = ∑ b in s, ∫⁻ a in t b, f a ∂μ := by simp only [← Finset.mem_coe, lintegral_bUnion₀ s.countable_to_set hm hd, ← s.tsum_subtype'] -#align measure_theory.lintegral_bUnion_finset₀ MeasureTheory.lintegral_bUnion_finset₀ - -theorem lintegral_bUnion_finset {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t) +#align measure_theory.lintegral_bUnion_finset₀ MeasureTheory.lintegral_biUnion_finset₀ + +/- warning: measure_theory.lintegral_bUnion_finset -> MeasureTheory.lintegral_biUnion_finset is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Finset.{u2} β} {t : β -> (Set.{u1} α)}, (Set.PairwiseDisjoint.{u1, u2} (Set.{u1} α) β (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} β) (Set.{u2} β) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} β) (Set.{u2} β) (Finset.Set.hasCoeT.{u2} β))) s) t) -> (forall (b : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) -> (MeasurableSet.{u1} α m (t b))) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (b : β) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) (fun (H : Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) b s) => t b)))) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal β (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (t b)) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Finset.{u2} β} {t : β -> (Set.{u1} α)}, (Set.PairwiseDisjoint.{u1, u2} (Set.{u1} α) β (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (Finset.toSet.{u2} β s) t) -> (forall (b : β), (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) -> (MeasurableSet.{u1} α m (t b))) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (b : β) => Set.iUnion.{u1, 0} α (Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) (fun (H : Membership.mem.{u2, u2} β (Finset.{u2} β) (Finset.instMembershipFinset.{u2} β) b s) => t b)))) (fun (a : α) => f a)) (Finset.sum.{0, u2} ENNReal β (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (b : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (t b)) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_bUnion_finset MeasureTheory.lintegral_biUnion_finsetₓ'. -/ +theorem lintegral_biUnion_finset {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t) (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ) = ∑ b in s, ∫⁻ a in t b, f a ∂μ := - lintegral_bUnion_finset₀ hd.AEDisjoint (fun b hb => (hm b hb).NullMeasurableSet) f -#align measure_theory.lintegral_bUnion_finset MeasureTheory.lintegral_bUnion_finset - + lintegral_biUnion_finset₀ hd.AEDisjoint (fun b hb => (hm b hb).NullMeasurableSet) f +#align measure_theory.lintegral_bUnion_finset MeasureTheory.lintegral_biUnion_finset + +/- warning: measure_theory.lintegral_Union_le -> MeasureTheory.lintegral_iUnion_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] (s : β -> (Set.{u1} α)) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u2} β] (s : β -> (Set.{u1} α)) (f : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Set.iUnion.{u1, succ u2} α β (fun (i : β) => s i))) (fun (a : α) => f a)) (tsum.{0, u2} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal β (fun (i : β) => MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (s i)) (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_Union_le MeasureTheory.lintegral_iUnion_leₓ'. -/ theorem lintegral_iUnion_le [Countable β] (s : β → Set α) (f : α → ℝ≥0∞) : (∫⁻ a in ⋃ i, s i, f a ∂μ) ≤ ∑' i, ∫⁻ a in s i, f a ∂μ := by @@ -1300,21 +1916,45 @@ theorem lintegral_iUnion_le [Countable β] (s : β → Set α) (f : α → ℝ exact lintegral_mono' restrict_Union_le le_rfl #align measure_theory.lintegral_Union_le MeasureTheory.lintegral_iUnion_le +/- warning: measure_theory.lintegral_union -> MeasureTheory.lintegral_union is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {A : Set.{u1} α} {B : Set.{u1} α}, (MeasurableSet.{u1} α m B) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) A B) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) A B)) (fun (a : α) => f a)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ B) (fun (a : α) => f a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {A : Set.{u1} α} {B : Set.{u1} α}, (MeasurableSet.{u1} α m B) -> (Disjoint.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) A B) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) A B)) (fun (a : α) => f a)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (a : α) => f a)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ B) (fun (a : α) => f a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_union MeasureTheory.lintegral_unionₓ'. -/ theorem lintegral_union {f : α → ℝ≥0∞} {A B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B) : (∫⁻ a in A ∪ B, f a ∂μ) = (∫⁻ a in A, f a ∂μ) + ∫⁻ a in B, f a ∂μ := by rw [restrict_union hAB hB, lintegral_add_measure] #align measure_theory.lintegral_union MeasureTheory.lintegral_union +/- warning: measure_theory.lintegral_inter_add_diff -> MeasureTheory.lintegral_inter_add_diff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {B : Set.{u1} α} (f : α -> ENNReal) (A : Set.{u1} α), (MeasurableSet.{u1} α m B) -> (Eq.{1} ENNReal (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) A B)) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A B)) (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (x : α) => f x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {B : Set.{u1} α} (f : α -> ENNReal) (A : Set.{u1} α), (MeasurableSet.{u1} α m B) -> (Eq.{1} ENNReal (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) A B)) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) A B)) (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (x : α) => f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_inter_add_diff MeasureTheory.lintegral_inter_add_diffₓ'. -/ theorem lintegral_inter_add_diff {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B) : ((∫⁻ x in A ∩ B, f x ∂μ) + ∫⁻ x in A \ B, f x ∂μ) = ∫⁻ x in A, f x ∂μ := by rw [← lintegral_add_measure, restrict_inter_add_diff _ hB] #align measure_theory.lintegral_inter_add_diff MeasureTheory.lintegral_inter_add_diff +/- warning: measure_theory.lintegral_add_compl -> MeasureTheory.lintegral_add_compl is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {A : Set.{u1} α}, (MeasurableSet.{u1} α m A) -> (Eq.{1} ENNReal (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A)) (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (f : α -> ENNReal) {A : Set.{u1} α}, (MeasurableSet.{u1} α m A) -> (Eq.{1} ENNReal (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ A) (fun (x : α) => f x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) A)) (fun (x : α) => f x))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_add_compl MeasureTheory.lintegral_add_complₓ'. -/ theorem lintegral_add_compl (f : α → ℝ≥0∞) {A : Set α} (hA : MeasurableSet A) : ((∫⁻ x in A, f x ∂μ) + ∫⁻ x in Aᶜ, f x ∂μ) = ∫⁻ x, f x ∂μ := by rw [← lintegral_add_measure, measure.restrict_add_restrict_compl hA] #align measure_theory.lintegral_add_compl MeasureTheory.lintegral_add_compl +/- warning: measure_theory.lintegral_max -> MeasureTheory.lintegral_max is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => LinearOrder.max.{0} ENNReal (ConditionallyCompleteLinearOrder.toLinearOrder.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.completeLinearOrder))) (f x) (g x))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x)))) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (g x) (f x)))) (fun (x : α) => f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => Max.max.{0} ENNReal (CanonicallyLinearOrderedAddMonoid.toMax.{0} ENNReal ENNReal.instCanonicallyLinearOrderedAddMonoidENNReal) (f x) (g x))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x)))) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (g x) (f x)))) (fun (x : α) => f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_max MeasureTheory.lintegral_maxₓ'. -/ theorem lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) : (∫⁻ x, max (f x) (g x) ∂μ) = (∫⁻ x in { x | f x ≤ g x }, g x ∂μ) + ∫⁻ x in { x | g x < f x }, f x ∂μ := @@ -1326,6 +1966,12 @@ theorem lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measur exacts[ae_of_all _ fun x => max_eq_right, ae_of_all _ fun x hx => max_eq_left (not_le.1 hx).le] #align measure_theory.lintegral_max MeasureTheory.lintegral_max +/- warning: measure_theory.set_lintegral_max -> MeasureTheory.set_lintegral_max is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall (s : Set.{u1} α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => LinearOrder.max.{0} ENNReal (ConditionallyCompleteLinearOrder.toLinearOrder.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.completeLinearOrder))) (f x) (g x))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (g x))))) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (g x) (f x))))) (fun (x : α) => f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall (s : Set.{u1} α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => Max.max.{0} ENNReal (CanonicallyLinearOrderedAddMonoid.toMax.{0} ENNReal ENNReal.instCanonicallyLinearOrderedAddMonoidENNReal) (f x) (g x))) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (setOf.{u1} α (fun (x : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (g x))))) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s (setOf.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (g x) (f x))))) (fun (x : α) => f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_max MeasureTheory.set_lintegral_maxₓ'. -/ theorem set_lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (s : Set α) : (∫⁻ x in s, max (f x) (g x) ∂μ) = (∫⁻ x in s ∩ { x | f x ≤ g x }, g x ∂μ) + ∫⁻ x in s ∩ { x | g x < f x }, f x ∂μ := @@ -1334,6 +1980,7 @@ theorem set_lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Me exacts[measurableSet_lt hg hf, measurableSet_le hf hg] #align measure_theory.set_lintegral_max MeasureTheory.set_lintegral_max +#print MeasureTheory.lintegral_map /- theorem lintegral_map {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g) : (∫⁻ a, f a ∂map g μ) = ∫⁻ a, f (g a) ∂μ := by @@ -1342,7 +1989,9 @@ theorem lintegral_map {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α convert simple_func.lintegral_map _ hg ext1 x; simp only [eapprox_comp hf hg, coe_comp] #align measure_theory.lintegral_map MeasureTheory.lintegral_map +-/ +#print MeasureTheory.lintegral_map' /- theorem lintegral_map' {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : AEMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) : (∫⁻ a, f a ∂Measure.map g μ) = ∫⁻ a, f (g a) ∂μ := @@ -1358,7 +2007,14 @@ theorem lintegral_map' {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : _ = ∫⁻ a, f (g a) ∂μ := lintegral_congr_ae (ae_eq_comp hg hf.ae_eq_mk.symm) #align measure_theory.lintegral_map' MeasureTheory.lintegral_map' +-/ +/- warning: measure_theory.lintegral_map_le -> MeasureTheory.lintegral_map_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {mβ : MeasurableSpace.{u2} β} (f : β -> ENNReal) {g : α -> β}, (Measurable.{u1, u2} α β m mβ g) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u2} β mβ (MeasureTheory.Measure.map.{u1, u2} α β mβ m g μ) (fun (a : β) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (g a)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {mβ : MeasurableSpace.{u2} β} (f : β -> ENNReal) {g : α -> β}, (Measurable.{u1, u2} α β m mβ g) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u2} β mβ (MeasureTheory.Measure.map.{u1, u2} α β mβ m g μ) (fun (a : β) => f a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f (g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_map_le MeasureTheory.lintegral_map_leₓ'. -/ theorem lintegral_map_le {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : α → β} (hg : Measurable g) : (∫⁻ a, f a ∂Measure.map g μ) ≤ ∫⁻ a, f (g a) ∂μ := by @@ -1368,17 +2024,27 @@ theorem lintegral_map_le {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : exact le_iSup_of_le (fun x => h'i (g x)) (le_of_eq (lintegral_map hi hg)) #align measure_theory.lintegral_map_le MeasureTheory.lintegral_map_le +#print MeasureTheory.lintegral_comp /- theorem lintegral_comp [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g) : lintegral μ (f ∘ g) = ∫⁻ a, f a ∂map g μ := (lintegral_map hf hg).symm #align measure_theory.lintegral_comp MeasureTheory.lintegral_comp +-/ +#print MeasureTheory.set_lintegral_map /- theorem set_lintegral_map [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β} (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) : (∫⁻ y in s, f y ∂map g μ) = ∫⁻ x in g ⁻¹' s, f (g x) ∂μ := by rw [restrict_map hg hs, lintegral_map hf hg] #align measure_theory.set_lintegral_map MeasureTheory.set_lintegral_map +-/ +/- warning: measure_theory.lintegral_indicator_const_comp -> MeasureTheory.lintegral_indicator_const_comp is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {mβ : MeasurableSpace.{u2} β} {f : α -> β} {s : Set.{u2} β}, (Measurable.{u1, u2} α β m mβ f) -> (MeasurableSet.{u2} β mβ s) -> (forall (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u2, 0} β ENNReal ENNReal.hasZero s (fun (_x : β) => c) (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.preimage.{u1, u2} α β f s)))) +but is expected to have type + forall {α : Type.{u1}} {β : Type.{u2}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {mβ : MeasurableSpace.{u2} β} {f : α -> β} {s : Set.{u2} β}, (Measurable.{u1, u2} α β m mβ f) -> (MeasurableSet.{u2} β mβ s) -> (forall (c : ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => Set.indicator.{u2, 0} β ENNReal instENNRealZero s (fun (_x : β) => c) (f a))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.preimage.{u1, u2} α β f s)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_indicator_const_comp MeasureTheory.lintegral_indicator_const_compₓ'. -/ theorem lintegral_indicator_const_comp {mβ : MeasurableSpace β} {f : α → β} {s : Set β} (hf : Measurable f) (hs : MeasurableSet s) (c : ℝ≥0∞) : (∫⁻ a, s.indicator (fun _ => c) (f a) ∂μ) = c * μ (f ⁻¹' s) := by @@ -1386,6 +2052,7 @@ theorem lintegral_indicator_const_comp {mβ : MeasurableSpace β} {f : α → β measure.map_apply hf hs] #align measure_theory.lintegral_indicator_const_comp MeasureTheory.lintegral_indicator_const_comp +#print MeasurableEmbedding.lintegral_map /- /-- If `g : α → β` is a measurable embedding and `f : β → ℝ≥0∞` is any function (not necessarily measurable), then `∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ`. Compare with `lintegral_map` wich applies to any measurable `g : α → β` but requires that `f` is measurable as well. -/ @@ -1402,7 +2069,9 @@ theorem MeasurableEmbedding.lintegral_map [MeasurableSpace β] {g : α → β} refine' lintegral_mono_ae (hg.ae_map_iff.2 <| eventually_of_forall fun x => _) exact (extend_apply _ _ _ _).trans_le (hf₀ _) #align measurable_embedding.lintegral_map MeasurableEmbedding.lintegral_map +-/ +#print MeasureTheory.lintegral_map_equiv /- /-- The `lintegral` transforms appropriately under a measurable equivalence `g : α ≃ᵐ β`. (Compare `lintegral_map`, which applies to a wider class of functions `g : α → β`, but requires measurability of the function being integrated.) -/ @@ -1410,37 +2079,54 @@ theorem lintegral_map_equiv [MeasurableSpace β] (f : β → ℝ≥0∞) (g : α (∫⁻ a, f a ∂map g μ) = ∫⁻ a, f (g a) ∂μ := g.MeasurableEmbedding.lintegral_map f #align measure_theory.lintegral_map_equiv MeasureTheory.lintegral_map_equiv +-/ +#print MeasureTheory.MeasurePreserving.lintegral_comp /- theorem MeasurePreserving.lintegral_comp {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) {f : β → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f (g a) ∂μ) = ∫⁻ b, f b ∂ν := by rw [← hg.map_eq, lintegral_map hf hg.measurable] #align measure_theory.measure_preserving.lintegral_comp MeasureTheory.MeasurePreserving.lintegral_comp +-/ +#print MeasureTheory.MeasurePreserving.lintegral_comp_emb /- theorem MeasurePreserving.lintegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) : (∫⁻ a, f (g a) ∂μ) = ∫⁻ b, f b ∂ν := by rw [← hg.map_eq, hge.lintegral_map] #align measure_theory.measure_preserving.lintegral_comp_emb MeasureTheory.MeasurePreserving.lintegral_comp_emb +-/ +#print MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage /- theorem MeasurePreserving.set_lintegral_comp_preimage {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a in g ⁻¹' s, f (g a) ∂μ) = ∫⁻ b in s, f b ∂ν := by rw [← hg.map_eq, set_lintegral_map hs hf hg.measurable] #align measure_theory.measure_preserving.set_lintegral_comp_preimage MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage +-/ +#print MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb /- theorem MeasurePreserving.set_lintegral_comp_preimage_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) (s : Set β) : (∫⁻ a in g ⁻¹' s, f (g a) ∂μ) = ∫⁻ b in s, f b ∂ν := by rw [← hg.map_eq, hge.restrict_map, hge.lintegral_map] #align measure_theory.measure_preserving.set_lintegral_comp_preimage_emb MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb +-/ +#print MeasureTheory.MeasurePreserving.set_lintegral_comp_emb /- theorem MeasurePreserving.set_lintegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) (s : Set α) : (∫⁻ a in s, f (g a) ∂μ) = ∫⁻ b in g '' s, f b ∂ν := by rw [← hg.set_lintegral_comp_preimage_emb hge, preimage_image_eq _ hge.injective] #align measure_theory.measure_preserving.set_lintegral_comp_emb MeasureTheory.MeasurePreserving.set_lintegral_comp_emb +-/ section DiracAndCount +/- warning: measurable_space.top.measurable_singleton_class -> MeasurableSpace.Top.measurableSingletonClass is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}}, MeasurableSingletonClass.{u1} α (Top.top.{u1} (MeasurableSpace.{u1} α) (CompleteLattice.toHasTop.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.completeLattice.{u1} α))) +but is expected to have type + forall {α : Type.{u1}}, MeasurableSingletonClass.{u1} α (Top.top.{u1} (MeasurableSpace.{u1} α) (CompleteLattice.toTop.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instCompleteLatticeMeasurableSpace.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measurable_space.top.measurable_singleton_class MeasurableSpace.Top.measurableSingletonClassₓ'. -/ instance (priority := 10) MeasurableSpace.Top.measurableSingletonClass {α : Type _} : @MeasurableSingletonClass α (⊤ : MeasurableSpace α) where measurableSet_singleton i := MeasurableSpace.measurableSet_top @@ -1448,14 +2134,24 @@ instance (priority := 10) MeasurableSpace.Top.measurableSingletonClass {α : Typ variable [MeasurableSpace α] +#print MeasureTheory.lintegral_dirac' /- theorem lintegral_dirac' (a : α) {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f a ∂dirac a) = f a := by simp [lintegral_congr_ae (ae_eq_dirac' hf)] #align measure_theory.lintegral_dirac' MeasureTheory.lintegral_dirac' +-/ +#print MeasureTheory.lintegral_dirac /- theorem lintegral_dirac [MeasurableSingletonClass α] (a : α) (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂dirac a) = f a := by simp [lintegral_congr_ae (ae_eq_dirac f)] #align measure_theory.lintegral_dirac MeasureTheory.lintegral_dirac +-/ +/- warning: measure_theory.set_lintegral_dirac' -> MeasureTheory.set_lintegral_dirac' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {a : α} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (forall [_inst_2 : Decidable (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s)], Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.restrict.{u1} α _inst_1 (MeasureTheory.Measure.dirac.{u1} α _inst_1 a) s) (fun (x : α) => f x)) (ite.{1} ENNReal (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) _inst_2 (f a) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {a : α} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (forall {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (forall [_inst_2 : Decidable (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s)], Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.restrict.{u1} α _inst_1 (MeasureTheory.Measure.dirac.{u1} α _inst_1 a) s) (fun (x : α) => f x)) (ite.{1} ENNReal (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) _inst_2 (f a) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_dirac' MeasureTheory.set_lintegral_dirac'ₓ'. -/ theorem set_lintegral_dirac' {a : α} {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)] : (∫⁻ x in s, f x ∂Measure.dirac a) = if a ∈ s then f a else 0 := @@ -1467,6 +2163,12 @@ theorem set_lintegral_dirac' {a : α} {f : α → ℝ≥0∞} (hf : Measurable f · exact lintegral_zero_measure _ #align measure_theory.set_lintegral_dirac' MeasureTheory.set_lintegral_dirac' +/- warning: measure_theory.set_lintegral_dirac -> MeasureTheory.set_lintegral_dirac is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {a : α} (f : α -> ENNReal) (s : Set.{u1} α) [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] [_inst_3 : Decidable (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s)], Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.restrict.{u1} α _inst_1 (MeasureTheory.Measure.dirac.{u1} α _inst_1 a) s) (fun (x : α) => f x)) (ite.{1} ENNReal (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) _inst_3 (f a) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {a : α} (f : α -> ENNReal) (s : Set.{u1} α) [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] [_inst_3 : Decidable (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s)], Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.restrict.{u1} α _inst_1 (MeasureTheory.Measure.dirac.{u1} α _inst_1 a) s) (fun (x : α) => f x)) (ite.{1} ENNReal (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) _inst_3 (f a) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_dirac MeasureTheory.set_lintegral_diracₓ'. -/ theorem set_lintegral_dirac {a : α} (f : α → ℝ≥0∞) (s : Set α) [MeasurableSingletonClass α] [Decidable (a ∈ s)] : (∫⁻ x in s, f x ∂Measure.dirac a) = if a ∈ s then f a else 0 := by @@ -1476,6 +2178,12 @@ theorem set_lintegral_dirac {a : α} (f : α → ℝ≥0∞) (s : Set α) [Measu · exact lintegral_zero_measure _ #align measure_theory.set_lintegral_dirac MeasureTheory.set_lintegral_dirac +/- warning: measure_theory.lintegral_count' -> MeasureTheory.lintegral_count' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace α (fun (a : α) => f a))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace f) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal α (fun (a : α) => f a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_count' MeasureTheory.lintegral_count'ₓ'. -/ theorem lintegral_count' {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a, f a ∂count) = ∑' a, f a := by rw [count, lintegral_sum_measure] @@ -1483,6 +2191,12 @@ theorem lintegral_count' {f : α → ℝ≥0∞} (hf : Measurable f) : (∫⁻ a exact funext fun a => lintegral_dirac' a hf #align measure_theory.lintegral_count' MeasureTheory.lintegral_count' +/- warning: measure_theory.lintegral_count -> MeasureTheory.lintegral_count is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace α (fun (a : α) => f a)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal α (fun (a : α) => f a)) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_count MeasureTheory.lintegral_countₓ'. -/ theorem lintegral_count [MeasurableSingletonClass α] (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂count) = ∑' a, f a := by @@ -1491,10 +2205,22 @@ theorem lintegral_count [MeasurableSingletonClass α] (f : α → ℝ≥0∞) : exact funext fun a => lintegral_dirac a f #align measure_theory.lintegral_count MeasureTheory.lintegral_count +/- warning: ennreal.tsum_const_eq -> ENNReal.tsum_const_eq is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] (c : ENNReal), Eq.{1} ENNReal (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace α (fun (i : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) c (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) (MeasureTheory.Measure.count.{u1} α _inst_1) (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] (c : ENNReal), Eq.{1} ENNReal (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal α (fun (i : α) => c)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) c (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1)) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align ennreal.tsum_const_eq ENNReal.tsum_const_eqₓ'. -/ theorem ENNReal.tsum_const_eq [MeasurableSingletonClass α] (c : ℝ≥0∞) : (∑' i : α, c) = c * Measure.count (univ : Set α) := by rw [← lintegral_count, lintegral_const] #align ennreal.tsum_const_eq ENNReal.tsum_const_eq +/- warning: ennreal.count_const_le_le_of_tsum_le -> ENNReal.count_const_le_le_of_tsum_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] {a : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace a) -> (forall {c : ENNReal}, (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace α (fun (i : α) => a i)) c) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Ne.{1} ENNReal ε (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) (MeasureTheory.Measure.count.{u1} α _inst_1) (setOf.{u1} α (fun (i : α) => LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) ε (a i)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toHasDiv.{0} ENNReal ENNReal.divInvMonoid)) c ε)))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] {a : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal _inst_1 ENNReal.measurableSpace a) -> (forall {c : ENNReal}, (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal α (fun (i : α) => a i)) c) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Ne.{1} ENNReal ε (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1)) (setOf.{u1} α (fun (i : α) => LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) ε (a i)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toDiv.{0} ENNReal ENNReal.instDivInvMonoidENNReal)) c ε)))) +Case conversion may be inaccurate. Consider using '#align ennreal.count_const_le_le_of_tsum_le ENNReal.count_const_le_le_of_tsum_leₓ'. -/ /-- Markov's inequality for the counting measure with hypothesis using `tsum` in `ℝ≥0∞`. -/ theorem ENNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0∞} (a_mble : Measurable a) {c : ℝ≥0∞} (tsum_le_c : (∑' i, a i) ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) @@ -1505,6 +2231,12 @@ theorem ENNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : exact ENNReal.div_le_div tsum_le_c rfl.le #align ennreal.count_const_le_le_of_tsum_le ENNReal.count_const_le_le_of_tsum_le +/- warning: nnreal.count_const_le_le_of_tsum_le -> NNReal.count_const_le_le_of_tsum_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] {a : α -> NNReal}, (Measurable.{u1, 0} α NNReal _inst_1 NNReal.measurableSpace a) -> (Summable.{0, u1} NNReal α (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) NNReal.topologicalSpace a) -> (forall {c : NNReal}, (LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (tsum.{0, u1} NNReal (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) NNReal.topologicalSpace α (fun (i : α) => a i)) c) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) (MeasureTheory.Measure.count.{u1} α _inst_1) (setOf.{u1} α (fun (i : α) => LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) ε (a i)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toHasDiv.{0} ENNReal ENNReal.divInvMonoid)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) ε))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α _inst_1] {a : α -> NNReal}, (Measurable.{u1, 0} α NNReal _inst_1 NNReal.measurableSpace a) -> (Summable.{0, u1} NNReal α (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal instNNRealStrictOrderedSemiring)) NNReal.instTopologicalSpaceNNReal a) -> (forall {c : NNReal}, (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (tsum.{0, u1} NNReal (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal instNNRealStrictOrderedSemiring)) NNReal.instTopologicalSpaceNNReal α (fun (i : α) => a i)) c) -> (forall {ε : NNReal}, (Ne.{1} NNReal ε (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 (MeasureTheory.Measure.count.{u1} α _inst_1)) (setOf.{u1} α (fun (i : α) => LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) ε (a i)))) (HDiv.hDiv.{0, 0, 0} ENNReal ENNReal ENNReal (instHDiv.{0} ENNReal (DivInvMonoid.toDiv.{0} ENNReal ENNReal.instDivInvMonoidENNReal)) (ENNReal.some c) (ENNReal.some ε))))) +Case conversion may be inaccurate. Consider using '#align nnreal.count_const_le_le_of_tsum_le NNReal.count_const_le_le_of_tsum_leₓ'. -/ /-- Markov's inequality for counting measure with hypothesis using `tsum` in `ℝ≥0`. -/ theorem NNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0} (a_mble : Measurable a) (a_summable : Summable a) {c : ℝ≥0} (tsum_le_c : (∑' i, a i) ≤ c) @@ -1530,6 +2262,12 @@ section Countable -/ +/- warning: measure_theory.lintegral_countable' -> MeasureTheory.lintegral_countable' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace α (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Countable.{succ u1} α] [_inst_2 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal α (fun (a : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_countable' MeasureTheory.lintegral_countable'ₓ'. -/ theorem lintegral_countable' [Countable α] [MeasurableSingletonClass α] (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂μ) = ∑' a, f a * μ {a} := by @@ -1538,26 +2276,50 @@ theorem lintegral_countable' [Countable α] [MeasurableSingletonClass α] (f : rw [lintegral_smul_measure, lintegral_dirac, mul_comm] #align measure_theory.lintegral_countable' MeasureTheory.lintegral_countable' +/- warning: measure_theory.lintegral_singleton' -> MeasureTheory.lintegral_singleton' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a)) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (a : α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a)) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_singleton' MeasureTheory.lintegral_singleton'ₓ'. -/ theorem lintegral_singleton' {f : α → ℝ≥0∞} (hf : Measurable f) (a : α) : (∫⁻ x in {a}, f x ∂μ) = f a * μ {a} := by simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac' _ hf, mul_comm] #align measure_theory.lintegral_singleton' MeasureTheory.lintegral_singleton' +/- warning: measure_theory.lintegral_singleton -> MeasureTheory.lintegral_singleton is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal) (a : α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a)) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal) (a : α), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a)) (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_singleton MeasureTheory.lintegral_singletonₓ'. -/ theorem lintegral_singleton [MeasurableSingletonClass α] (f : α → ℝ≥0∞) (a : α) : (∫⁻ x in {a}, f x ∂μ) = f a * μ {a} := by simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac, mul_comm] #align measure_theory.lintegral_singleton MeasureTheory.lintegral_singleton +/- warning: measure_theory.lintegral_countable -> MeasureTheory.lintegral_countable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal) {s : Set.{u1} α}, (Set.Countable.{u1} α s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (fun (a : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) a)) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s))))) a)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (f : α -> ENNReal) {s : Set.{u1} α}, (Set.Countable.{u1} α s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (a : α) => f a)) (tsum.{0, u1} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal (Set.Elem.{u1} α s) (fun (a : Set.Elem.{u1} α s) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) a)) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) a)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_countable MeasureTheory.lintegral_countableₓ'. -/ theorem lintegral_countable [MeasurableSingletonClass α] (f : α → ℝ≥0∞) {s : Set α} (hs : s.Countable) : (∫⁻ a in s, f a ∂μ) = ∑' a : s, f a * μ {(a : α)} := calc (∫⁻ a in s, f a ∂μ) = ∫⁻ a in ⋃ x ∈ s, {x}, f a ∂μ := by rw [bUnion_of_singleton] _ = ∑' a : s, ∫⁻ x in {a}, f x ∂μ := - (lintegral_bUnion hs (fun _ _ => measurableSet_singleton _) (pairwise_disjoint_fiber id s) _) + (lintegral_biUnion hs (fun _ _ => measurableSet_singleton _) (pairwise_disjoint_fiber id s) _) _ = ∑' a : s, f a * μ {(a : α)} := by simp only [lintegral_singleton] #align measure_theory.lintegral_countable MeasureTheory.lintegral_countable +/- warning: measure_theory.lintegral_insert -> MeasureTheory.lintegral_insert is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] {a : α} {s : Set.{u1} α}, (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.hasInsert.{u1} α) a s)) (fun (x : α) => f x)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f a) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] {a : α} {s : Set.{u1} α}, (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s)) -> (forall (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) a s)) (fun (x : α) => f x)) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f a) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => f x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_insert MeasureTheory.lintegral_insertₓ'. -/ theorem lintegral_insert [MeasurableSingletonClass α] {a : α} {s : Set α} (h : a ∉ s) (f : α → ℝ≥0∞) : (∫⁻ x in insert a s, f x ∂μ) = f a * μ {a} + ∫⁻ x in s, f x ∂μ := by @@ -1566,16 +2328,34 @@ theorem lintegral_insert [MeasurableSingletonClass α] {a : α} {s : Set α} (h rwa [disjoint_singleton_right] #align measure_theory.lintegral_insert MeasureTheory.lintegral_insert +/- warning: measure_theory.lintegral_finset -> MeasureTheory.lintegral_finset is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (s : Finset.{u1} α) (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} α) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (Finset.Set.hasCoeT.{u1} α))) s)) (fun (x : α) => f x)) (Finset.sum.{0, u1} ENNReal α (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) s (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f x) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] (s : Finset.{u1} α) (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ (Finset.toSet.{u1} α s)) (fun (x : α) => f x)) (Finset.sum.{0, u1} ENNReal α (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) s (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f x) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_finset MeasureTheory.lintegral_finsetₓ'. -/ theorem lintegral_finset [MeasurableSingletonClass α] (s : Finset α) (f : α → ℝ≥0∞) : (∫⁻ x in s, f x ∂μ) = ∑ x in s, f x * μ {x} := by simp only [lintegral_countable _ s.countable_to_set, ← s.tsum_subtype'] #align measure_theory.lintegral_finset MeasureTheory.lintegral_finset +/- warning: measure_theory.lintegral_fintype -> MeasureTheory.lintegral_fintype is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] [_inst_2 : Fintype.{u1} α] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Finset.sum.{0, u1} ENNReal α (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) (Finset.univ.{u1} α _inst_2) (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f x) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : MeasurableSingletonClass.{u1} α m] [_inst_2 : Fintype.{u1} α] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Finset.sum.{0, u1} ENNReal α (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) (Finset.univ.{u1} α _inst_2) (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f x) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_fintype MeasureTheory.lintegral_fintypeₓ'. -/ theorem lintegral_fintype [MeasurableSingletonClass α] [Fintype α] (f : α → ℝ≥0∞) : (∫⁻ x, f x ∂μ) = ∑ x, f x * μ {x} := by rw [← lintegral_finset, Finset.coe_univ, measure.restrict_univ] #align measure_theory.lintegral_fintype MeasureTheory.lintegral_fintype +/- warning: measure_theory.lintegral_unique -> MeasureTheory.lintegral_unique is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Unique.{succ u1} α] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (f (Inhabited.default.{succ u1} α (Unique.inhabited.{succ u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : Unique.{succ u1} α] (f : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (f (Inhabited.default.{succ u1} α (Unique.instInhabited.{succ u1} α _inst_1))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_unique MeasureTheory.lintegral_uniqueₓ'. -/ theorem lintegral_unique [Unique α] (f : α → ℝ≥0∞) : (∫⁻ x, f x ∂μ) = f default * μ univ := calc (∫⁻ x, f x ∂μ) = ∫⁻ x, f default ∂μ := lintegral_congr <| Unique.forall_iff.2 rfl @@ -1585,6 +2365,12 @@ theorem lintegral_unique [Unique α] (f : α → ℝ≥0∞) : (∫⁻ x, f x end Countable +/- warning: measure_theory.ae_lt_top -> MeasureTheory.ae_lt_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_lt_top MeasureTheory.ae_lt_topₓ'. -/ theorem ae_lt_top {f : α → ℝ≥0∞} (hf : Measurable f) (h2f : (∫⁻ x, f x ∂μ) ≠ ∞) : ∀ᵐ x ∂μ, f x < ∞ := by simp_rw [ae_iff, ENNReal.not_lt_top] @@ -1598,12 +2384,24 @@ theorem ae_lt_top {f : α → ℝ≥0∞} (hf : Measurable f) (h2f : (∫⁻ x, simp [ENNReal.top_mul', preimage, h] #align measure_theory.ae_lt_top MeasureTheory.ae_lt_top +/- warning: measure_theory.ae_lt_top' -> MeasureTheory.ae_lt_top' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_lt_top' MeasureTheory.ae_lt_top'ₓ'. -/ theorem ae_lt_top' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h2f : (∫⁻ x, f x ∂μ) ≠ ∞) : ∀ᵐ x ∂μ, f x < ∞ := haveI h2f_meas : (∫⁻ x, hf.mk f x ∂μ) ≠ ∞ := by rwa [← lintegral_congr_ae hf.ae_eq_mk] (ae_lt_top hf.measurable_mk h2f_meas).mp (hf.ae_eq_mk.mono fun x hx h => by rwa [hx]) #align measure_theory.ae_lt_top' MeasureTheory.ae_lt_top' +/- warning: measure_theory.set_lintegral_lt_top_of_bdd_above -> MeasureTheory.set_lintegral_lt_top_of_bddAbove is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (BddAbove.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring))) (Set.image.{u1, 0} α NNReal f s)) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {s : Set.{u1} α}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (BddAbove.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring)) (Set.image.{u1, 0} α NNReal f s)) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => ENNReal.some (f x))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_lt_top_of_bdd_above MeasureTheory.set_lintegral_lt_top_of_bddAboveₓ'. -/ theorem set_lintegral_lt_top_of_bddAbove {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0} (hf : Measurable f) (hbdd : BddAbove (f '' s)) : (∫⁻ x in s, f x ∂μ) < ∞ := by @@ -1617,12 +2415,24 @@ theorem set_lintegral_lt_top_of_bddAbove {s : Set α} (hs : μ s ≠ ∞) {f : simp [hs] #align measure_theory.set_lintegral_lt_top_of_bdd_above MeasureTheory.set_lintegral_lt_top_of_bddAbove +/- warning: measure_theory.set_lintegral_lt_top_of_is_compact -> MeasureTheory.set_lintegral_lt_top_of_isCompact is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : OpensMeasurableSpace.{u1} α _inst_1 m] {s : Set.{u1} α}, (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (IsCompact.{u1} α _inst_1 s) -> (forall {f : α -> NNReal}, (Continuous.{u1, 0} α NNReal _inst_1 NNReal.topologicalSpace f) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : OpensMeasurableSpace.{u1} α _inst_1 m] {s : Set.{u1} α}, (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (IsCompact.{u1} α _inst_1 s) -> (forall {f : α -> NNReal}, (Continuous.{u1, 0} α NNReal _inst_1 NNReal.instTopologicalSpaceNNReal f) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => ENNReal.some (f x))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_lt_top_of_is_compact MeasureTheory.set_lintegral_lt_top_of_isCompactₓ'. -/ theorem set_lintegral_lt_top_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α] {s : Set α} (hs : μ s ≠ ∞) (hsc : IsCompact s) {f : α → ℝ≥0} (hf : Continuous f) : (∫⁻ x in s, f x ∂μ) < ∞ := set_lintegral_lt_top_of_bddAbove hs hf.Measurable (hsc.image hf).BddAbove #align measure_theory.set_lintegral_lt_top_of_is_compact MeasureTheory.set_lintegral_lt_top_of_isCompact +/- warning: is_finite_measure.lintegral_lt_top_of_bounded_to_ennreal -> IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNReal is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [μ_fin : MeasureTheory.FiniteMeasure.{u1} α _inst_1 μ] {f : α -> ENNReal}, (Exists.{1} NNReal (fun (c : NNReal) => forall (x : α), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) c))) -> (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α _inst_1 μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [μ_fin : MeasureTheory.FiniteMeasure.{u1} α _inst_1 μ] {f : α -> ENNReal}, (Exists.{1} NNReal (fun (c : NNReal) => forall (x : α), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (ENNReal.some c))) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α _inst_1 μ (fun (x : α) => f x)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) +Case conversion may be inaccurate. Consider using '#align is_finite_measure.lintegral_lt_top_of_bounded_to_ennreal IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNRealₓ'. -/ theorem IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNReal {α : Type _} [MeasurableSpace α] (μ : Measure α) [μ_fin : FiniteMeasure μ] {f : α → ℝ≥0∞} (f_bdd : ∃ c : ℝ≥0, ∀ x, f x ≤ c) : (∫⁻ x, f x ∂μ) < ∞ := by @@ -1632,26 +2442,33 @@ theorem IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNReal {α : Type _} [Me exact ENNReal.mul_lt_top ennreal.coe_lt_top.ne μ_fin.measure_univ_lt_top.ne #align is_finite_measure.lintegral_lt_top_of_bounded_to_ennreal IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNReal +#print MeasureTheory.Measure.withDensity /- /-- Given a measure `μ : measure α` and a function `f : α → ℝ≥0∞`, `μ.with_density f` is the measure such that for a measurable set `s` we have `μ.with_density f s = ∫⁻ a in s, f a ∂μ`. -/ def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : Measure α := Measure.ofMeasurable (fun s hs => ∫⁻ a in s, f a ∂μ) (by simp) fun s hs hd => lintegral_iUnion hs hd _ #align measure_theory.measure.with_density MeasureTheory.Measure.withDensity +-/ +#print MeasureTheory.withDensity_apply /- @[simp] theorem withDensity_apply (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : μ.withDensity f s = ∫⁻ a in s, f a ∂μ := Measure.ofMeasurable_apply s hs #align measure_theory.with_density_apply MeasureTheory.withDensity_apply +-/ +#print MeasureTheory.withDensity_congr_ae /- theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : μ.withDensity f = μ.withDensity g := by apply measure.ext fun s hs => _ rw [with_density_apply _ hs, with_density_apply _ hs] exact lintegral_congr_ae (ae_restrict_of_ae h) #align measure_theory.with_density_congr_ae MeasureTheory.withDensity_congr_ae +-/ +#print MeasureTheory.withDensity_add_left /- theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) : μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by @@ -1660,26 +2477,34 @@ theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α ← lintegral_add_left hf] rfl #align measure_theory.with_density_add_left MeasureTheory.withDensity_add_left +-/ +#print MeasureTheory.withDensity_add_right /- theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) : μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by simpa only [add_comm] using with_density_add_left hg f #align measure_theory.with_density_add_right MeasureTheory.withDensity_add_right +-/ +#print MeasureTheory.withDensity_add_measure /- theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) : (μ + ν).withDensity f = μ.withDensity f + ν.withDensity f := by ext1 s hs simp only [with_density_apply f hs, restrict_add, lintegral_add_measure, measure.add_apply] #align measure_theory.with_density_add_measure MeasureTheory.withDensity_add_measure +-/ +#print MeasureTheory.withDensity_sum /- theorem withDensity_sum {ι : Type _} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) : (Sum μ).withDensity f = Sum fun n => (μ n).withDensity f := by ext1 s hs simp_rw [sum_apply _ hs, with_density_apply f hs, restrict_sum μ hs, lintegral_sum_measure] #align measure_theory.with_density_sum MeasureTheory.withDensity_sum +-/ +#print MeasureTheory.withDensity_smul /- theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) : μ.withDensity (r • f) = r • μ.withDensity f := by @@ -1688,7 +2513,14 @@ theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurab smul_eq_mul, ← lintegral_const_mul r hf] rfl #align measure_theory.with_density_smul MeasureTheory.withDensity_smul +-/ +/- warning: measure_theory.with_density_smul' -> MeasureTheory.withDensity_smul' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{succ u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.withDensity.{u1} α m μ (SMul.smul.{0, u1} ENNReal (α -> ENNReal) (Function.hasSMul.{u1, 0, 0} α ENNReal ENNReal (Mul.toSMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) r f)) (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) m) r (MeasureTheory.Measure.withDensity.{u1} α m μ f))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} (r : ENNReal) (f : α -> ENNReal), (Ne.{1} ENNReal r (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{succ u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.withDensity.{u1} α m μ (HSMul.hSMul.{0, u1, u1} ENNReal (α -> ENNReal) (α -> ENNReal) (instHSMul.{0, u1} ENNReal (α -> ENNReal) (Pi.instSMul.{u1, 0, 0} α ENNReal (fun (a._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.34021 : α) => ENNReal) (fun (i : α) => Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))))) r f)) (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m)) r (MeasureTheory.Measure.withDensity.{u1} α m μ f))) +Case conversion may be inaccurate. Consider using '#align measure_theory.with_density_smul' MeasureTheory.withDensity_smul'ₓ'. -/ theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : μ.withDensity (r • f) = r • μ.withDensity f := by @@ -1698,6 +2530,12 @@ theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ rfl #align measure_theory.with_density_smul' MeasureTheory.withDensity_smul' +/- warning: measure_theory.is_finite_measure_with_density -> MeasureTheory.finiteMeasure_withDensity is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.FiniteMeasure.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (Ne.{1} ENNReal (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => f a)) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.FiniteMeasure.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f)) +Case conversion may be inaccurate. Consider using '#align measure_theory.is_finite_measure_with_density MeasureTheory.finiteMeasure_withDensityₓ'. -/ theorem finiteMeasure_withDensity {f : α → ℝ≥0∞} (hf : (∫⁻ a, f a ∂μ) ≠ ∞) : FiniteMeasure (μ.withDensity f) := { @@ -1705,6 +2543,7 @@ theorem finiteMeasure_withDensity {f : α → ℝ≥0∞} (hf : (∫⁻ a, f a rwa [with_density_apply _ MeasurableSet.univ, measure.restrict_univ, lt_top_iff_ne_top] } #align measure_theory.is_finite_measure_with_density MeasureTheory.finiteMeasure_withDensity +#print MeasureTheory.withDensity_absolutelyContinuous /- theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : μ.withDensity f ≪ μ := by @@ -1712,21 +2551,27 @@ theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure rw [with_density_apply _ hs₁] exact set_lintegral_measure_zero _ _ hs₂ #align measure_theory.with_density_absolutely_continuous MeasureTheory.withDensity_absolutelyContinuous +-/ +#print MeasureTheory.withDensity_zero /- @[simp] theorem withDensity_zero : μ.withDensity 0 = 0 := by ext1 s hs simp [with_density_apply _ hs] #align measure_theory.with_density_zero MeasureTheory.withDensity_zero +-/ +#print MeasureTheory.withDensity_one /- @[simp] theorem withDensity_one : μ.withDensity 1 = μ := by ext1 s hs simp [with_density_apply _ hs] #align measure_theory.with_density_one MeasureTheory.withDensity_one +-/ +#print MeasureTheory.withDensity_tsum /- theorem withDensity_tsum {f : ℕ → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) : μ.withDensity (∑' n, f n) = Sum fun n => μ.withDensity (f n) := by @@ -1736,7 +2581,9 @@ theorem withDensity_tsum {f : ℕ → α → ℝ≥0∞} (h : ∀ i, Measurable rw [← lintegral_tsum fun i => (h i).AEMeasurable] refine' lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable) #align measure_theory.with_density_tsum MeasureTheory.withDensity_tsum +-/ +#print MeasureTheory.withDensity_indicator /- theorem withDensity_indicator {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) : μ.withDensity (s.indicator f) = (μ.restrict s).withDensity f := by @@ -1744,12 +2591,16 @@ theorem withDensity_indicator {s : Set α} (hs : MeasurableSet s) (f : α → rw [with_density_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, ← with_density_apply _ ht] #align measure_theory.with_density_indicator MeasureTheory.withDensity_indicator +-/ +#print MeasureTheory.withDensity_indicator_one /- theorem withDensity_indicator_one {s : Set α} (hs : MeasurableSet s) : μ.withDensity (s.indicator 1) = μ.restrict s := by rw [with_density_indicator hs, with_density_one] #align measure_theory.with_density_indicator_one MeasureTheory.withDensity_indicator_one +-/ +#print MeasureTheory.withDensity_ofReal_mutuallySingular /- theorem withDensity_ofReal_mutuallySingular {f : α → ℝ} (hf : Measurable f) : (μ.withDensity fun x => ENNReal.ofReal <| f x) ⟂ₘ μ.withDensity fun x => ENNReal.ofReal <| -f x := @@ -1764,7 +2615,9 @@ theorem withDensity_ofReal_mutuallySingular {f : α → ℝ} (hf : Measurable f) (ae_restrict_mem hS.compl).mono fun x hx => ENNReal.ofReal_eq_zero.2 (not_lt.1 <| mt neg_pos.1 hx) #align measure_theory.with_density_of_real_mutually_singular MeasureTheory.withDensity_ofReal_mutuallySingular +-/ +#print MeasureTheory.restrict_withDensity /- theorem restrict_withDensity {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) : (μ.withDensity f).restrict s = (μ.restrict s).withDensity f := by @@ -1772,13 +2625,26 @@ theorem restrict_withDensity {s : Set α} (hs : MeasurableSet s) (f : α → ℝ rw [restrict_apply ht, with_density_apply _ ht, with_density_apply _ (ht.inter hs), restrict_restrict ht] #align measure_theory.restrict_with_density MeasureTheory.restrict_withDensity +-/ +/- warning: measure_theory.with_density_eq_zero -> MeasureTheory.withDensity_eq_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{succ u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.withDensity.{u1} α m μ f) (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m))))) -> (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (OfNat.mk.{u1} (α -> ENNReal) 0 (Zero.zero.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => ENNReal.hasZero)))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Eq.{succ u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.withDensity.{u1} α m μ f) (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α m) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instZero.{u1} α m)))) -> (Filter.EventuallyEq.{u1, 0} α ENNReal (MeasureTheory.Measure.ae.{u1} α m μ) f (OfNat.ofNat.{u1} (α -> ENNReal) 0 (Zero.toOfNat0.{u1} (α -> ENNReal) (Pi.instZero.{u1, 0} α (fun (a._@.Mathlib.Order.Filter.Basic._hyg.19136 : α) => ENNReal) (fun (i : α) => instENNRealZero))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.with_density_eq_zero MeasureTheory.withDensity_eq_zeroₓ'. -/ theorem withDensity_eq_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h : μ.withDensity f = 0) : f =ᵐ[μ] 0 := by rw [← lintegral_eq_zero_iff' hf, ← set_lintegral_univ, ← with_density_apply _ MeasurableSet.univ, h, measure.coe_zero, Pi.zero_apply] #align measure_theory.with_density_eq_zero MeasureTheory.withDensity_eq_zero +/- warning: measure_theory.with_density_apply_eq_zero -> MeasureTheory.withDensity_apply_eq_zero is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {s : Set.{u1} α}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) (MeasureTheory.Measure.withDensity.{u1} α m μ f) s) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) (setOf.{u1} α (fun (x : α) => Ne.{1} ENNReal (f x) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))))) s)) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal} {s : Set.{u1} α}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f)) s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (setOf.{u1} α (fun (x : α) => Ne.{1} ENNReal (f x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)))) s)) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.with_density_apply_eq_zero MeasureTheory.withDensity_apply_eq_zeroₓ'. -/ theorem withDensity_apply_eq_zero {f : α → ℝ≥0∞} {s : Set α} (hf : Measurable f) : μ.withDensity f s = 0 ↔ μ ({ x | f x ≠ 0 } ∩ s) = 0 := by @@ -1814,6 +2680,12 @@ theorem withDensity_apply_eq_zero {f : α → ℝ≥0∞} {s : Set α} (hf : Mea simp only [imp_self, Pi.zero_apply, imp_true_iff] #align measure_theory.with_density_apply_eq_zero MeasureTheory.withDensity_apply_eq_zero +/- warning: measure_theory.ae_with_density_iff -> MeasureTheory.ae_withDensity_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : α -> Prop} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Filter.Eventually.{u1} α (fun (x : α) => p x) (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f))) (Filter.Eventually.{u1} α (fun (x : α) => (Ne.{1} ENNReal (f x) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (p x)) (MeasureTheory.Measure.ae.{u1} α m μ))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : α -> Prop} {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Iff (Filter.Eventually.{u1} α (fun (x : α) => p x) (MeasureTheory.Measure.ae.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f))) (Filter.Eventually.{u1} α (fun (x : α) => (Ne.{1} ENNReal (f x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (p x)) (MeasureTheory.Measure.ae.{u1} α m μ))) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_with_density_iff MeasureTheory.ae_withDensity_iffₓ'. -/ theorem ae_withDensity_iff {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f) : (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ, f x ≠ 0 → p x := by @@ -1823,6 +2695,7 @@ theorem ae_withDensity_iff {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measu simp only [exists_prop, mem_inter_iff, iff_self_iff, mem_set_of_eq, not_forall] #align measure_theory.ae_with_density_iff MeasureTheory.ae_withDensity_iff +#print MeasureTheory.ae_withDensity_iff_ae_restrict /- theorem ae_withDensity_iff_ae_restrict {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f) : (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ.restrict { x | f x ≠ 0 }, p x := by @@ -1830,7 +2703,14 @@ theorem ae_withDensity_iff_ae_restrict {p : α → Prop} {f : α → ℝ≥0∞} · rfl · exact hf (measurable_set_singleton 0).compl #align measure_theory.ae_with_density_iff_ae_restrict MeasureTheory.ae_withDensity_iff_ae_restrict +-/ +/- warning: measure_theory.ae_measurable_with_density_ennreal_iff -> MeasureTheory.aEMeasurable_withDensity_eNNReal_iff is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> ENNReal}, Iff (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x)))) (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x)) (g x)) μ)) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> NNReal}, (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace f) -> (forall {g : α -> ENNReal}, Iff (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g (MeasureTheory.Measure.withDensity.{u1} α m μ (fun (x : α) => ENNReal.some (f x)))) (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m (fun (x : α) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (ENNReal.some (f x)) (g x)) μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.ae_measurable_with_density_ennreal_iff MeasureTheory.aEMeasurable_withDensity_eNNReal_iffₓ'. -/ theorem aEMeasurable_withDensity_eNNReal_iff {f : α → ℝ≥0} (hf : Measurable f) {g : α → ℝ≥0∞} : AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔ AEMeasurable (fun x => (f x : ℝ≥0∞) * g x) μ := @@ -1866,6 +2746,12 @@ variable {m m0 : MeasurableSpace α} include m +/- warning: measure_theory.lintegral_with_density_eq_lintegral_mul -> MeasureTheory.lintegral_withDensity_eq_lintegral_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_eq_lintegral_mul MeasureTheory.lintegral_withDensity_eq_lintegral_mulₓ'. -/ /-- This is Exercise 1.2.1 from [tao2010]. It allows you to express integration of a measurable function with respect to `(μ.with_density f)` as an integral with respect to `μ`, called the base measure. `μ` is often the Lebesgue measure, and in this circumstance `f` is the probability density @@ -1879,7 +2765,7 @@ theorem lintegral_withDensity_eq_lintegral_mul (μ : Measure α) {f : α → ℝ (h_mf : Measurable f) : ∀ {g : α → ℝ≥0∞}, Measurable g → (∫⁻ a, g a ∂μ.withDensity f) = ∫⁻ a, (f * g) a ∂μ := by - apply Measurable.eNNReal_induction + apply Measurable.ennreal_induction · intro c s h_ms simp [*, mul_comm _ c, ← indicator_mul_right] · intro g h h_univ h_mea_g h_mea_h h_ind_g h_ind_h @@ -1889,12 +2775,24 @@ theorem lintegral_withDensity_eq_lintegral_mul (μ : Measure α) {f : α → ℝ simp [lintegral_supr, ENNReal.mul_iSup, h_mf.mul (h_mea_g _), *] #align measure_theory.lintegral_with_density_eq_lintegral_mul MeasureTheory.lintegral_withDensity_eq_lintegral_mul +/- warning: measure_theory.set_lintegral_with_density_eq_set_lintegral_mul -> MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) s) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g x)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal} {g : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace g) -> (forall {s : Set.{u1} α}, (MeasurableSet.{u1} α m s) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) s) (fun (x : α) => g x)) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.restrict.{u1} α m μ s) (fun (x : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g x)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.set_lintegral_with_density_eq_set_lintegral_mul MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mulₓ'. -/ theorem set_lintegral_withDensity_eq_set_lintegral_mul (μ : Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) {s : Set α} (hs : MeasurableSet s) : (∫⁻ x in s, g x ∂μ.withDensity f) = ∫⁻ x in s, (f * g) x ∂μ := by rw [restrict_with_density hs, lintegral_with_density_eq_lintegral_mul _ hf hg] #align measure_theory.set_lintegral_with_density_eq_set_lintegral_mul MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul +/- warning: measure_theory.lintegral_with_density_eq_lintegral_mul₀' -> MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g (MeasureTheory.Measure.withDensity.{u1} α m μ f)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g (MeasureTheory.Measure.withDensity.{u1} α m μ f)) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_eq_lintegral_mul₀' MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀'ₓ'. -/ /-- The Lebesgue integral of `g` with respect to the measure `μ.with_density f` coincides with the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a version without conditions on `g` but requiring that `f` is almost everywhere finite, see @@ -1933,12 +2831,24 @@ theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → #align measure_theory.lintegral_with_density_eq_lintegral_mul₀' MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀' +/- warning: measure_theory.lintegral_with_density_eq_lintegral_mul₀ -> MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (forall {g : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m g μ) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_eq_lintegral_mul₀ MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀ₓ'. -/ theorem lintegral_withDensity_eq_lintegral_mul₀ {μ : Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) : (∫⁻ a, g a ∂μ.withDensity f) = ∫⁻ a, (f * g) a ∂μ := lintegral_withDensity_eq_lintegral_mul₀' hf (hg.mono' (withDensity_absolutelyContinuous μ f)) #align measure_theory.lintegral_with_density_eq_lintegral_mul₀ MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀ +/- warning: measure_theory.lintegral_with_density_le_lintegral_mul -> MeasureTheory.lintegral_withDensity_le_lintegral_mul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (g : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (forall (g : α -> ENNReal), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_le_lintegral_mul MeasureTheory.lintegral_withDensity_le_lintegral_mulₓ'. -/ theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by @@ -1949,6 +2859,12 @@ theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ exact le_iSup_of_le A (le_of_eq (lintegral_with_density_eq_lintegral_mul _ f_meas i_meas)) #align measure_theory.lintegral_with_density_le_lintegral_mul MeasureTheory.lintegral_withDensity_le_lintegral_mul +/- warning: measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable -> MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m ENNReal.measurableSpace f) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurableₓ'. -/ theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) = ∫⁻ a, (f * g) a ∂μ := @@ -1976,13 +2892,21 @@ theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) rw [← mul_assoc, ENNReal.mul_inv_cancel hx h'x.ne, one_mul] #align measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable +#print MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable /- theorem set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ.restrict s, f x < ∞) : (∫⁻ a in s, g a ∂μ.withDensity f) = ∫⁻ a in s, (f * g) a ∂μ := by rw [restrict_with_density hs, lintegral_with_density_eq_lintegral_mul_non_measurable _ f_meas hf] #align measure_theory.set_lintegral_with_density_eq_set_lintegral_mul_non_measurable MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable +-/ +/- warning: measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable₀ -> MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀ is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => Distrib.toHasMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) f g a))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m f μ) -> (Filter.Eventually.{u1} α (fun (x : α) => LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f x) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (MeasureTheory.Measure.ae.{u1} α m μ)) -> (forall (g : α -> ENNReal), Eq.{1} ENNReal (MeasureTheory.lintegral.{u1} α m (MeasureTheory.Measure.withDensity.{u1} α m μ f) (fun (a : α) => g a)) (MeasureTheory.lintegral.{u1} α m μ (fun (a : α) => HMul.hMul.{u1, u1, u1} (α -> ENNReal) (α -> ENNReal) (α -> ENNReal) (instHMul.{u1} (α -> ENNReal) (Pi.instMul.{u1, 0} α (fun (ᾰ : α) => ENNReal) (fun (i : α) => CanonicallyOrderedCommSemiring.toMul.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) f g a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable₀ MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀ₓ'. -/ theorem lintegral_withDensity_eq_lintegral_mul_non_measurable₀ (μ : Measure α) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h'f : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) = ∫⁻ a, (f * g) a ∂μ := @@ -2005,13 +2929,16 @@ theorem lintegral_withDensity_eq_lintegral_mul_non_measurable₀ (μ : Measure #align measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable₀ MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀ +#print MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ /- theorem set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ (μ : Measure α) {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞) (hs : MeasurableSet s) (h'f : ∀ᵐ x ∂μ.restrict s, f x < ∞) : (∫⁻ a in s, g a ∂μ.withDensity f) = ∫⁻ a in s, (f * g) a ∂μ := by rw [restrict_with_density hs, lintegral_with_density_eq_lintegral_mul_non_measurable₀ _ hf h'f] #align measure_theory.set_lintegral_with_density_eq_set_lintegral_mul_non_measurable₀ MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ +-/ +#print MeasureTheory.withDensity_mul /- theorem withDensity_mul (μ : Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) : μ.withDensity (f * g) = (μ.withDensity f).withDensity g := by @@ -2019,7 +2946,14 @@ theorem withDensity_mul (μ : Measure α) {f g : α → ℝ≥0∞} (hf : Measur simp [with_density_apply _ hs, restrict_with_density hs, lintegral_with_density_eq_lintegral_mul _ hf hg] #align measure_theory.with_density_mul MeasureTheory.withDensity_mul +-/ +/- warning: measure_theory.exists_pos_lintegral_lt_of_sigma_finite -> MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) [_inst_1 : MeasureTheory.SigmaFinite.{u1} α m μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (α -> NNReal) (fun (g : α -> NNReal) => And (forall (x : α), LT.lt.{0} NNReal (Preorder.toHasLt.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (OfNat.ofNat.{0} NNReal 0 (OfNat.mk.{0} NNReal 0 (Zero.zero.{0} NNReal (MulZeroClass.toHasZero.{0} NNReal (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring))))))) (g x)) (And (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace g) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (g x))) ε)))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} (μ : MeasureTheory.Measure.{u1} α m) [_inst_1 : MeasureTheory.SigmaFinite.{u1} α m μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (α -> NNReal) (fun (g : α -> NNReal) => And (forall (x : α), LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (OfNat.ofNat.{0} NNReal 0 (Zero.toOfNat0.{0} NNReal instNNRealZero)) (g x)) (And (Measurable.{u1, 0} α NNReal m NNReal.measurableSpace g) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (g x))) ε)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_pos_lintegral_lt_of_sigma_finite MeasureTheory.exists_pos_lintegral_lt_of_sigmaFiniteₓ'. -/ /-- In a sigma-finite measure space, there exists an integrable function which is positive everywhere (and with an arbitrarily small integral). -/ theorem exists_pos_lintegral_lt_of_sigmaFinite (μ : Measure α) [SigmaFinite μ] {ε : ℝ≥0∞} @@ -2041,11 +2975,12 @@ theorem exists_pos_lintegral_lt_of_sigmaFinite (μ : Measure α) [SigmaFinite μ measurable_spanning_sets_index, mul_comm] using δsum #align measure_theory.exists_pos_lintegral_lt_of_sigma_finite MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite +#print MeasureTheory.lintegral_trim /- theorem lintegral_trim {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : measurable[m] f) : (∫⁻ a, f a ∂μ.trim hm) = ∫⁻ a, f a ∂μ := by refine' - @Measurable.eNNReal_induction α m (fun f => (∫⁻ a, f a ∂μ.trim hm) = ∫⁻ a, f a ∂μ) _ _ _ f hf + @Measurable.ennreal_induction α m (fun f => (∫⁻ a, f a ∂μ.trim hm) = ∫⁻ a, f a ∂μ) _ _ _ f hf · intro c s hs rw [lintegral_indicator _ hs, lintegral_indicator _ (hm s hs), set_lintegral_const, set_lintegral_const] @@ -2062,17 +2997,26 @@ theorem lintegral_trim {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} congr exact funext fun n => hf_prop n #align measure_theory.lintegral_trim MeasureTheory.lintegral_trim +-/ +#print MeasureTheory.lintegral_trim_ae /- theorem lintegral_trim_ae {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.trim hm)) : (∫⁻ a, f a ∂μ.trim hm) = ∫⁻ a, f a ∂μ := by rw [lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), lintegral_congr_ae hf.ae_eq_mk, lintegral_trim hm hf.measurable_mk] #align measure_theory.lintegral_trim_ae MeasureTheory.lintegral_trim_ae +-/ section SigmaFinite variable {E : Type _} [NormedAddCommGroup E] [MeasurableSpace E] [OpensMeasurableSpace E] +/- warning: measure_theory.univ_le_of_forall_fin_meas_le -> MeasureTheory.univ_le_of_forall_fin_meas_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : (Set.{u1} α) -> ENNReal}, (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m0) (fun (_x : MeasureTheory.Measure.{u1} α m0) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m0) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f s) C)) -> (forall (S : Nat -> (Set.{u1} α)), (forall (n : Nat), MeasurableSet.{u1} α m (S n)) -> (Monotone.{0, u1} Nat (Set.{u1} α) (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α))))))) S) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => S n))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) Nat (fun (n : Nat) => f (S n))))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (f (Set.univ.{u1} α)) C) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instLEMeasurableSpace.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : (Set.{u1} α) -> ENNReal}, (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m0 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f s) C)) -> (forall (S : Nat -> (Set.{u1} α)), (forall (n : Nat), MeasurableSet.{u1} α m (S n)) -> (Monotone.{0, u1} Nat (Set.{u1} α) (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α))))))) S) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f (Set.iUnion.{u1, 1} α Nat (fun (n : Nat) => S n))) (iSup.{0, 1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) Nat (fun (n : Nat) => f (S n))))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (f (Set.univ.{u1} α)) C) +Case conversion may be inaccurate. Consider using '#align measure_theory.univ_le_of_forall_fin_meas_le MeasureTheory.univ_le_of_forall_fin_meas_leₓ'. -/ theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, measurable_set[m] s → μ s ≠ ∞ → f s ≤ C) (h_F_lim : @@ -2087,6 +3031,12 @@ theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFi exact ((le_trim hm).trans_lt (@measure_spanning_sets_lt_top _ m (μ.trim hm) _ n)).Ne #align measure_theory.univ_le_of_forall_fin_meas_le MeasureTheory.univ_le_of_forall_fin_meas_le +/- warning: measure_theory.lintegral_le_of_forall_fin_meas_le_of_measurable -> MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m0 ENNReal.measurableSpace f) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m0) (fun (_x : MeasureTheory.Measure.{u1} α m0) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m0) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m0 μ (fun (x : α) => f x)) C) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instLEMeasurableSpace.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : α -> ENNReal}, (Measurable.{u1, 0} α ENNReal m0 ENNReal.measurableSpace f) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m0 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m0 μ (fun (x : α) => f x)) C) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_le_of_forall_fin_meas_le_of_measurable MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurableₓ'. -/ /-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral over the whole space is bounded by that same constant. Version for a measurable function. @@ -2130,6 +3080,12 @@ theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : · exact le_rfl #align measure_theory.lintegral_le_of_forall_fin_meas_le_of_measurable MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable +/- warning: measure_theory.lintegral_le_of_forall_fin_meas_le' -> MeasureTheory.lintegral_le_of_forall_fin_meas_le' is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.hasLe.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m0 f μ) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m0) (fun (_x : MeasureTheory.Measure.{u1} α m0) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m0) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m0 μ (fun (x : α) => f x)) C) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {m0 : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m0} (hm : LE.le.{u1} (MeasurableSpace.{u1} α) (MeasurableSpace.instLEMeasurableSpace.{u1} α) m m0) [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m (MeasureTheory.Measure.trim.{u1} α m m0 μ hm)] (C : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace m0 f μ) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α m s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m0 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m0 (MeasureTheory.Measure.restrict.{u1} α m0 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m0 μ (fun (x : α) => f x)) C) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_le_of_forall_fin_meas_le' MeasureTheory.lintegral_le_of_forall_fin_meas_le'ₓ'. -/ /-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral over the whole space is bounded by that same constant. -/ @@ -2149,6 +3105,12 @@ theorem lintegral_le_of_forall_fin_meas_le' {μ : Measure α} (hm : m ≤ m0) [S omit m +/- warning: measure_theory.lintegral_le_of_forall_fin_meas_le -> MeasureTheory.lintegral_le_of_forall_fin_meas_le is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_4 : MeasurableSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_4} [_inst_5 : MeasureTheory.SigmaFinite.{u1} α _inst_4 μ] (C : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace _inst_4 f μ) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α _inst_4 s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_4) (fun (_x : MeasureTheory.Measure.{u1} α _inst_4) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_4) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α _inst_4 (MeasureTheory.Measure.restrict.{u1} α _inst_4 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α _inst_4 μ (fun (x : α) => f x)) C) +but is expected to have type + forall {α : Type.{u1}} [_inst_4 : MeasurableSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_4} [_inst_5 : MeasureTheory.SigmaFinite.{u1} α _inst_4 μ] (C : ENNReal) {f : α -> ENNReal}, (AEMeasurable.{u1, 0} α ENNReal ENNReal.measurableSpace _inst_4 f μ) -> (forall (s : Set.{u1} α), (MeasurableSet.{u1} α _inst_4 s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_4 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α _inst_4 (MeasureTheory.Measure.restrict.{u1} α _inst_4 μ s) (fun (x : α) => f x)) C)) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α _inst_4 μ (fun (x : α) => f x)) C) +Case conversion may be inaccurate. Consider using '#align measure_theory.lintegral_le_of_forall_fin_meas_le MeasureTheory.lintegral_le_of_forall_fin_meas_leₓ'. -/ /-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure and the measure is σ-finite, then the integral over the whole space is bounded by that same constant. -/ @@ -2161,6 +3123,12 @@ theorem lintegral_le_of_forall_fin_meas_le [MeasurableSpace α] {μ : Measure α -- mathport name: «expr →ₛ » local infixr:25 " →ₛ " => SimpleFunc +/- warning: measure_theory.simple_func.exists_lt_lintegral_simple_func_of_lt_lintegral -> MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m μ] {f : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal} {L : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) f x)))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (g : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) f x)) (And (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x))))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m μ] {f : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal} {L : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal f x)))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (g : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal f x)) (And (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x))))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.simple_func.exists_lt_lintegral_simple_func_of_lt_lintegral MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegralₓ'. -/ theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} {μ : Measure α} [SigmaFinite μ] {f : α →ₛ ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) : ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ (∫⁻ x, g x ∂μ) < ∞ ∧ L < ∫⁻ x, g x ∂μ := @@ -2218,6 +3186,12 @@ theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : Measurabl exact le_rfl #align measure_theory.simple_func.exists_lt_lintegral_simple_func_of_lt_lintegral MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral +/- warning: measure_theory.exists_lt_lintegral_simple_func_of_lt_lintegral -> MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m μ] {f : α -> NNReal} {L : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (f x)))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (g : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} NNReal (Preorder.toHasLe.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (OrderedCancelAddCommMonoid.toPartialOrder.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x) (f x)) (And (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x))) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => (fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (coeFn.{succ u1, succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (_x : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => α -> NNReal) (MeasureTheory.SimpleFunc.instCoeFun.{u1, 0} α NNReal m) g x))))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} [_inst_4 : MeasureTheory.SigmaFinite.{u1} α m μ] {f : α -> NNReal} {L : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (f x)))) -> (Exists.{succ u1} (MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) (fun (g : MeasureTheory.SimpleFunc.{u1, 0} α m NNReal) => And (forall (x : α), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal (StrictOrderedSemiring.toPartialOrder.{0} NNReal instNNRealStrictOrderedSemiring))) (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x) (f x)) (And (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x))) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) L (MeasureTheory.lintegral.{u1} α m μ (fun (x : α) => ENNReal.some (MeasureTheory.SimpleFunc.toFun.{u1, 0} α m NNReal g x))))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.exists_lt_lintegral_simple_func_of_lt_lintegral MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegralₓ'. -/ theorem exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} {μ : Measure α} [SigmaFinite μ] {f : α → ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) : ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ (∫⁻ x, g x ∂μ) < ∞ ∧ L < ∫⁻ x, g x ∂μ := @@ -2232,6 +3206,7 @@ theorem exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} exact ⟨g, fun x => (hg x).trans (coe_le_coe.1 (hg₀ x)), gL, gtop⟩ #align measure_theory.exists_lt_lintegral_simple_func_of_lt_lintegral MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral +#print MeasureTheory.exists_absolutelyContinuous_finiteMeasure /- /-- A sigma-finite measure is absolutely continuous with respect to some finite measure. -/ theorem exists_absolutelyContinuous_finiteMeasure {m : MeasurableSpace α} (μ : Measure α) [SigmaFinite μ] : ∃ ν : Measure α, FiniteMeasure ν ∧ μ ≪ ν := @@ -2251,6 +3226,7 @@ theorem exists_absolutelyContinuous_finiteMeasure {m : MeasurableSpace α} (μ : conv_lhs => rw [this] exact with_density_absolutely_continuous _ _ #align measure_theory.exists_absolutely_continuous_is_finite_measure MeasureTheory.exists_absolutelyContinuous_finiteMeasure +-/ end SigmaFinite diff --git a/Mathbin/MeasureTheory/Measure/Regular.lean b/Mathbin/MeasureTheory/Measure/Regular.lean index e3564992e7..d3c3662b71 100644 --- a/Mathbin/MeasureTheory/Measure/Regular.lean +++ b/Mathbin/MeasureTheory/Measure/Regular.lean @@ -141,6 +141,7 @@ namespace MeasureTheory namespace Measure /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/ +#print MeasureTheory.Measure.InnerRegular /- /-- We say that a measure `μ` is *inner regular* with respect to predicates `p q : set α → Prop`, if for every `U` such that `q U` and `r < μ U`, there exists a subset `K ⊆ U` satisfying `p K` of measure greater than `r`. @@ -150,12 +151,19 @@ repeating the proofs. -/ def InnerRegular {α} {m : MeasurableSpace α} (μ : Measure α) (p q : Set α → Prop) := ∀ ⦃U⦄, q U → ∀ r < μ U, ∃ (K : _)(_ : K ⊆ U), p K ∧ r < μ K #align measure_theory.measure.inner_regular MeasureTheory.Measure.InnerRegular +-/ namespace InnerRegular variable {α : Type _} {m : MeasurableSpace α} {μ : Measure α} {p q : Set α → Prop} {U : Set α} {ε : ℝ≥0∞} +/- warning: measure_theory.measure.inner_regular.measure_eq_supr -> MeasureTheory.Measure.InnerRegular.measure_eq_iSup is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (q U) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (p K) (fun (hK : p K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ K))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (q U) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (p K) (fun (hK : p K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) K))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSupₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) : μ U = ⨆ (K) (_ : K ⊆ U) (hK : p K), μ K := @@ -165,6 +173,12 @@ theorem measure_eq_iSup (H : InnerRegular μ p q) (hU : q U) : simpa only [lt_iSup_iff, exists_prop] using H hU r hr #align measure_theory.measure.inner_regular.measure_eq_supr MeasureTheory.Measure.InnerRegular.measure_eq_iSup +/- warning: measure_theory.measure.inner_regular.exists_subset_lt_add -> MeasureTheory.Measure.InnerRegular.exists_subset_lt_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α} {ε : ENNReal}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) -> (q U) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ U) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => And (p K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α m) (fun (_x : MeasureTheory.Measure.{u1} α m) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α m) μ K) ε))))) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop} {U : Set.{u1} α} {ε : ENNReal}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) -> (q U) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) U) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (And (p K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α m μ) K) ε))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_addₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞) (hε : ε ≠ 0) : ∃ (K : _)(_ : K ⊆ U), p K ∧ μ U < μ K + ε := @@ -176,6 +190,12 @@ theorem exists_subset_lt_add (H : InnerRegular μ p q) (h0 : p ∅) (hU : q U) ( exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩ #align measure_theory.measure.inner_regular.exists_subset_lt_add MeasureTheory.Measure.InnerRegular.exists_subset_lt_add +/- warning: measure_theory.measure.inner_regular.map -> MeasureTheory.Measure.InnerRegular.map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : MeasurableSpace.{u2} β] {μ : MeasureTheory.Measure.{u1} α _inst_1} {pa : (Set.{u1} α) -> Prop} {qa : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ pa qa) -> (forall (f : Equiv.{succ u1, succ u2} α β), (AEMeasurable.{u1, u2} α β _inst_2 _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) μ) -> (forall {pb : (Set.{u2} β) -> Prop} {qb : (Set.{u2} β) -> Prop}, (forall (U : Set.{u2} β), (qb U) -> (qa (Set.preimage.{u1, u2} α β (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) U))) -> (forall (K : Set.{u1} α), (pa K) -> (pb (Set.image.{u1, u2} α β (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) K))) -> (forall (K : Set.{u2} β), (pb K) -> (MeasurableSet.{u2} β _inst_2 K)) -> (forall (U : Set.{u2} β), (qb U) -> (MeasurableSet.{u2} β _inst_2 U)) -> (MeasureTheory.Measure.InnerRegular.{u2} β _inst_2 (MeasureTheory.Measure.map.{u1, u2} α β _inst_2 _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} α β) (fun (_x : Equiv.{succ u1, succ u2} α β) => α -> β) (Equiv.hasCoeToFun.{succ u1, succ u2} α β) f) μ) pb qb))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : MeasurableSpace.{u1} β] {μ : MeasureTheory.Measure.{u2} α _inst_1} {pa : (Set.{u2} α) -> Prop} {qa : (Set.{u2} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u2} α _inst_1 μ pa qa) -> (forall (f : Equiv.{succ u2, succ u1} α β), (AEMeasurable.{u2, u1} α β _inst_2 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) μ) -> (forall {pb : (Set.{u1} β) -> Prop} {qb : (Set.{u1} β) -> Prop}, (forall (U : Set.{u1} β), (qb U) -> (qa (Set.preimage.{u2, u1} α β (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) U))) -> (forall (K : Set.{u2} α), (pa K) -> (pb (Set.image.{u2, u1} α β (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) K))) -> (forall (K : Set.{u1} β), (pb K) -> (MeasurableSet.{u1} β _inst_2 K)) -> (forall (U : Set.{u1} β), (qb U) -> (MeasurableSet.{u1} β _inst_2 U)) -> (MeasureTheory.Measure.InnerRegular.{u1} β _inst_2 (MeasureTheory.Measure.map.{u2, u1} α β _inst_2 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => β) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} α β) f) μ) pb qb))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.mapₓ'. -/ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop} (H : InnerRegular μ pa qa) (f : α ≃ β) (hf : AEMeasurable f μ) {pb qb : Set β → Prop} (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K)) @@ -188,6 +208,12 @@ theorem map {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} rwa [map_apply_of_ae_measurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image] #align measure_theory.measure.inner_regular.map MeasureTheory.Measure.InnerRegular.map +/- warning: measure_theory.measure.inner_regular.smul -> MeasureTheory.Measure.InnerRegular.smul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (forall (c : ENNReal), MeasureTheory.Measure.InnerRegular.{u1} α m (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) m) c μ) p q) +but is expected to have type + forall {α : Type.{u1}} {m : MeasurableSpace.{u1} α} {μ : MeasureTheory.Measure.{u1} α m} {p : (Set.{u1} α) -> Prop} {q : (Set.{u1} α) -> Prop}, (MeasureTheory.Measure.InnerRegular.{u1} α m μ p q) -> (forall (c : ENNReal), MeasureTheory.Measure.InnerRegular.{u1} α m (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.{u1} α m) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α m) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) m)) c μ) p q) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smulₓ'. -/ theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ) p q := by intro U hU r hr @@ -195,17 +221,20 @@ theorem smul (H : InnerRegular μ p q) (c : ℝ≥0∞) : InnerRegular (c • μ simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr #align measure_theory.measure.inner_regular.smul MeasureTheory.Measure.InnerRegular.smul +#print MeasureTheory.Measure.InnerRegular.trans /- theorem trans {q' : Set α → Prop} (H : InnerRegular μ p q) (H' : InnerRegular μ q q') : InnerRegular μ p q' := by intro U hU r hr rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩ exact ⟨K, hKF.trans hFU, hpK, hrK⟩ #align measure_theory.measure.inner_regular.trans MeasureTheory.Measure.InnerRegular.trans +-/ end InnerRegular variable {α β : Type _} [MeasurableSpace α] [TopologicalSpace α] {μ : Measure α} +#print MeasureTheory.Measure.OuterRegular /- /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/ /-- A measure `μ` is outer regular if `μ(A) = inf {μ(U) | A ⊆ U open}` for a measurable set `A`. @@ -216,7 +245,9 @@ class OuterRegular (μ : Measure α) : Prop where OuterRegular : ∀ ⦃A : Set α⦄, MeasurableSet A → ∀ r > μ A, ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < r #align measure_theory.measure.outer_regular MeasureTheory.Measure.OuterRegular +-/ +#print MeasureTheory.Measure.Regular /- /-- A measure `μ` is regular if - it is finite on all compact sets; - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable; @@ -226,7 +257,9 @@ class OuterRegular (μ : Measure α) : Prop where class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular μ : Prop where InnerRegular : InnerRegular μ IsCompact IsOpen #align measure_theory.measure.regular MeasureTheory.Measure.Regular +-/ +#print MeasureTheory.Measure.WeaklyRegular /- /-- A measure `μ` is weakly regular if - it is outer regular: `μ(A) = inf {μ(U) | A ⊆ U open}` for `A` measurable; - it is inner regular for open sets, using closed sets: @@ -235,7 +268,9 @@ class Regular (μ : Measure α) extends FiniteMeasureOnCompacts μ, OuterRegular class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where InnerRegular : InnerRegular μ IsClosed IsOpen #align measure_theory.measure.weakly_regular MeasureTheory.Measure.WeaklyRegular +-/ +#print MeasureTheory.Measure.Regular.weaklyRegular /- -- see Note [lower instance priority] /-- A regular measure is weakly regular. -/ instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : WeaklyRegular μ @@ -243,13 +278,22 @@ instance (priority := 100) Regular.weaklyRegular [T2Space α] [Regular μ] : Wea let ⟨K, hKU, hcK, hK⟩ := Regular.innerRegular hU r hr ⟨K, hKU, hcK.IsClosed, hK⟩ #align measure_theory.measure.regular.weakly_regular MeasureTheory.Measure.Regular.weaklyRegular +-/ namespace OuterRegular +#print MeasureTheory.Measure.OuterRegular.zero /- instance zero : OuterRegular (0 : Measure α) := ⟨fun A hA r hr => ⟨univ, subset_univ A, isOpen_univ, hr⟩⟩ #align measure_theory.measure.outer_regular.zero MeasureTheory.Measure.OuterRegular.zero +-/ +/- warning: set.exists_is_open_lt_of_lt -> Set.exists_isOpen_lt_of_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α) (r : ENNReal), (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) r) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 U) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) r)))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α) (r : ENNReal), (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) r) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 U) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) r)))) +Case conversion may be inaccurate. Consider using '#align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_ltₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/ /-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with measure less than `r`. -/ @@ -262,6 +306,12 @@ theorem Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0 exact ⟨U, (subset_to_measurable _ _).trans hAU, hUo, hU⟩ #align set.exists_is_open_lt_of_lt Set.exists_isOpen_lt_of_lt +/- warning: set.measure_eq_infi_is_open -> Set.measure_eq_iInf_isOpen is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (iInf.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (U : Set.{u1} α) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) A U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) A U) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasInf.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsOpen.{u1} α _inst_2 U) (fun (h2 : IsOpen.{u1} α _inst_2 U) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U)))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (iInf.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (U : Set.{u1} α) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) A U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) A U) => iInf.{0, 0} ENNReal (ConditionallyCompleteLattice.toInfSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsOpen.{u1} α _inst_2 U) (fun (h2 : IsOpen.{u1} α _inst_2 U) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U)))) +Case conversion may be inaccurate. Consider using '#align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpenₓ'. -/ /-- For an outer regular measure, the measure of a set is the infimum of the measures of open sets containing it. -/ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] : @@ -272,12 +322,24 @@ theorem Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular simpa only [iInf_lt_iff, exists_prop] using A.exists_is_open_lt_of_lt r hr #align set.measure_eq_infi_is_open Set.measure_eq_iInf_isOpen +/- warning: set.exists_is_open_lt_add -> Set.exists_isOpen_lt_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 U) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) ε)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 U) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) ε)))))) +Case conversion may be inaccurate. Consider using '#align set.exists_is_open_lt_add Set.exists_isOpen_lt_addₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/ theorem Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U < μ A + ε := A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε) #align set.exists_is_open_lt_add Set.exists_isOpen_lt_add +/- warning: set.exists_is_open_le_add -> Set.exists_isOpen_le_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 U) (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) ε))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (A : Set.{u1} α) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 U) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) ε))))) +Case conversion may be inaccurate. Consider using '#align set.exists_is_open_le_add Set.exists_isOpen_le_addₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ (U : _)(_ : U ⊇ A), IsOpen U ∧ μ U ≤ μ A + ε := @@ -288,6 +350,12 @@ theorem Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ exact ⟨U, AU, U_open, hU.le⟩ #align set.exists_is_open_le_add Set.exists_isOpen_le_add +/- warning: measurable_set.exists_is_open_diff_lt -> MeasurableSet.exists_isOpen_diff_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {A : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) (fun (H : Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) U A) => And (IsOpen.{u1} α _inst_2 U) (And (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) U A)) ε)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {A : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (Superset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) U A) (And (IsOpen.{u1} α _inst_2 U) (And (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) U A)) ε)))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_ltₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A) -/ theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : @@ -298,6 +366,12 @@ theorem MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α} (hA : exact measure_diff_lt_of_lt_add hA hAU hA' hU #align measurable_set.exists_is_open_diff_lt MeasurableSet.exists_isOpen_diff_lt +/- warning: measure_theory.measure.outer_regular.map -> MeasureTheory.Measure.OuterRegular.map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : BorelSpace.{u2} β _inst_5 _inst_4] (f : Homeomorph.{u1, u2} α β _inst_2 _inst_5) (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_7 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.OuterRegular.{u2} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u1, u2} α β _inst_4 _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Homeomorph.{u1, u2} α β _inst_2 _inst_5) (fun (_x : Homeomorph.{u1, u2} α β _inst_2 _inst_5) => α -> β) (Homeomorph.hasCoeToFun.{u1, u2} α β _inst_2 _inst_5) f) μ) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OpensMeasurableSpace.{u2} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : BorelSpace.{u1} β _inst_5 _inst_4] (f : Homeomorph.{u2, u1} α β _inst_2 _inst_5) (μ : MeasureTheory.Measure.{u2} α _inst_1) [_inst_7 : MeasureTheory.Measure.OuterRegular.{u2} α _inst_1 _inst_2 μ], MeasureTheory.Measure.OuterRegular.{u1} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u2, u1} α β _inst_4 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α (fun (_x : α) => β) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (Homeomorph.instEquivLikeHomeomorph.{u2, u1} α β _inst_2 _inst_5))) f) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.mapₓ'. -/ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] : (Measure.map f μ).OuterRegular := by @@ -309,6 +383,12 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica rwa [map_apply f.measurable this.measurable_set, f.preimage_symm, f.preimage_image] #align measure_theory.measure.outer_regular.map MeasureTheory.Measure.OuterRegular.map +/- warning: measure_theory.measure.outer_regular.smul -> MeasureTheory.Measure.OuterRegular.smul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) _inst_1) x μ)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.{u1} α _inst_1) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) _inst_1)) x μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.outer_regular.smul MeasureTheory.Measure.OuterRegular.smulₓ'. -/ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).OuterRegular := by rcases eq_or_ne x 0 with (rfl | h0) @@ -322,6 +402,7 @@ protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx end OuterRegular /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » A n) -/ +#print MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular /- /-- If a measure `μ` admits finite spanning open sets such that the restriction of `μ` to each set is outer regular, then the original measure is outer regular as well. -/ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α} @@ -363,11 +444,18 @@ protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] { _ < r := hδε #align measure_theory.measure.finite_spanning_sets_in.outer_regular MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular +-/ namespace InnerRegular variable {p q : Set α → Prop} {U s : Set α} {ε r : ℝ≥0∞} +/- warning: measure_theory.measure.inner_regular.measurable_set_of_open -> MeasureTheory.Measure.InnerRegular.measurableSet_of_open is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} {p : (Set.{u1} α) -> Prop} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (IsOpen.{u1} α _inst_2)) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α))) -> (forall {{s : Set.{u1} α}} {{U : Set.{u1} α}}, (p s) -> (IsOpen.{u1} α _inst_2 U) -> (p (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s U))) -> (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} {p : (Set.{u1} α) -> Prop} [_inst_3 : MeasureTheory.Measure.OuterRegular.{u1} α _inst_1 _inst_2 μ], (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (IsOpen.{u1} α _inst_2)) -> (p (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α))) -> (forall {{s : Set.{u1} α}} {{U : Set.{u1} α}}, (p s) -> (IsOpen.{u1} α _inst_2 U) -> (p (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s U))) -> (MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ p (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.inner_regular.measurable_set_of_open MeasureTheory.Measure.InnerRegular.measurableSet_of_openₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/ /-- If a measure is inner regular (using closed or compact sets), then every measurable set of finite measure can by approximated by a (closed or compact) subset. -/ @@ -401,6 +489,7 @@ open Finset /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (ε «expr ≠ » 0) -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » s) -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U «expr ⊇ » s) -/ +#print MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite /- /-- In a finite measure space, assume that any open set can be approximated from inside by closed sets. Then the measure is weakly regular. -/ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure μ] @@ -475,7 +564,9 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [FiniteMeasure _ ≤ μ (⋃ n, s n) + ε := add_le_add_left (hδε.le.trans ENNReal.half_le_self) _ #align measure_theory.measure.inner_regular.weakly_regular_of_finite MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite +-/ +#print MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace /- /-- In a metric space (or even a pseudo emetric space), an open set can be approximated from inside by closed sets. -/ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpace X] @@ -487,7 +578,9 @@ theorem of_pseudoEMetricSpace {X : Type _} [PseudoEMetricSpace X] [MeasurableSpa rcases lt_iSup_iff.1 hr with ⟨n, hn⟩ exact ⟨F n, subset_Union _ _, F_closed n, hn⟩ #align measure_theory.measure.inner_regular.of_pseudo_emetric_space MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace +-/ +#print MeasureTheory.Measure.InnerRegular.isCompact_isClosed /- /-- In a `σ`-compact space, any closed set can be approximated by a compact subset. -/ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace X] [MeasurableSpace X] (μ : Measure X) : InnerRegular μ IsCompact IsClosed := @@ -504,15 +597,24 @@ theorem isCompact_isClosed {X : Type _} [TopologicalSpace X] [SigmaCompactSpace rcases lt_iSup_iff.1 hr with ⟨n, hn⟩ exact ⟨_, inter_subset_left _ _, hBc n, hn⟩ #align measure_theory.measure.inner_regular.is_compact_is_closed MeasureTheory.Measure.InnerRegular.isCompact_isClosed +-/ end InnerRegular namespace Regular +#print MeasureTheory.Measure.Regular.zero /- instance zero : Regular (0 : Measure α) := ⟨fun U hU r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩ #align measure_theory.measure.regular.zero MeasureTheory.Measure.Regular.zero +-/ +/- warning: is_open.exists_lt_is_compact -> IsOpen.exists_lt_isCompact is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))) +Case conversion may be inaccurate. Consider using '#align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompactₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » U) -/ /-- If `μ` is a regular measure, then any open set can be approximated by a compact subset. -/ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞} @@ -520,17 +622,35 @@ theorem IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) Regular.innerRegular hU r hr #align is_open.exists_lt_is_compact IsOpen.exists_lt_isCompact +/- warning: is_open.measure_eq_supr_is_compact -> IsOpen.measure_eq_iSup_isCompact is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsCompact.{u1} α _inst_2 K) (fun (h2 : IsCompact.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) (fun (h : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsCompact.{u1} α _inst_2 K) (fun (h2 : IsCompact.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))) +Case conversion may be inaccurate. Consider using '#align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompactₓ'. -/ /-- The measure of an open set is the supremum of the measures of compact sets it contains. -/ theorem IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) [Regular μ] : μ U = ⨆ (K : Set α) (h : K ⊆ U) (h2 : IsCompact K), μ K := Regular.innerRegular.measure_eq_iSup hU #align is_open.measure_eq_supr_is_compact IsOpen.measure_eq_iSup_isCompact +/- warning: measure_theory.measure.regular.exists_compact_not_null -> MeasureTheory.Measure.Regular.exists_compact_not_null is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Iff (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (IsCompact.{u1} α _inst_2 K) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))))) (Ne.{succ u1} (MeasureTheory.Measure.{u1} α _inst_1) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (OfNat.mk.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.zero.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], Iff (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (IsCompact.{u1} α _inst_2 K) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))))) (Ne.{succ u1} (MeasureTheory.Measure.{u1} α _inst_1) μ (OfNat.ofNat.{u1} (MeasureTheory.Measure.{u1} α _inst_1) 0 (Zero.toOfNat0.{u1} (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instZero.{u1} α _inst_1)))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_nullₓ'. -/ theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by simp_rw [Ne.def, ← measure_univ_eq_zero, is_open_univ.measure_eq_supr_is_compact, ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff] #align measure_theory.measure.regular.exists_compact_not_null MeasureTheory.Measure.Regular.exists_compact_not_null +/- warning: measure_theory.measure.regular.inner_regular_measurable -> MeasureTheory.Measure.Regular.innerRegular_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsCompact.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsCompact.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurableₓ'. -/ /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a compact subset. See also `measurable_set.exists_is_compact_lt_add` and `measurable_set.exists_lt_is_compact_of_ne_top`. -/ @@ -539,6 +659,12 @@ theorem innerRegular_measurable [Regular μ] : Regular.innerRegular.measurableSet_of_open isCompact_empty fun _ _ => IsCompact.diff #align measure_theory.measure.regular.inner_regular_measurable MeasureTheory.Measure.Regular.innerRegular_measurable +/- warning: measurable_set.exists_is_compact_lt_add -> MeasurableSet.exists_isCompact_lt_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) ε)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) ε)))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_addₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a compact subset. See also `measurable_set.exists_lt_is_compact_of_ne_top`. -/ @@ -547,6 +673,12 @@ theorem MeasurableSet.exists_isCompact_lt_add [Regular μ] ⦃A : Set α⦄ (hA Regular.innerRegular_measurable.exists_subset_lt_add isCompact_empty ⟨hA, h'A⟩ h'A hε #align measurable_set.exists_is_compact_lt_add MeasurableSet.exists_isCompact_lt_add +/- warning: measurable_set.exists_is_compact_diff_lt -> MeasurableSet.exists_isCompact_diff_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : T2Space.{u1} α _inst_2] [_inst_5 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A K)) ε))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : T2Space.{u1} α _inst_2] [_inst_5 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) A K)) ε))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_ltₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a compact subset. See also `measurable_set.exists_is_compact_lt_add` and @@ -562,6 +694,12 @@ theorem MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Spac hK⟩ #align measurable_set.exists_is_compact_diff_lt MeasurableSet.exists_isCompact_diff_lt +/- warning: measurable_set.exists_lt_is_compact_of_ne_top -> MeasurableSet.exists_lt_isCompact_of_ne_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsCompact.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_topₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a compact subset. See also `measurable_set.exists_is_compact_lt_add`. -/ @@ -570,6 +708,12 @@ theorem MeasurableSet.exists_lt_isCompact_of_ne_top [Regular μ] ⦃A : Set α Regular.innerRegular_measurable ⟨hA, h'A⟩ _ hr #align measurable_set.exists_lt_is_compact_of_ne_top MeasurableSet.exists_lt_isCompact_of_ne_top +/- warning: measurable_set.measure_eq_supr_is_compact_of_ne_top -> MeasurableSet.measure_eq_iSup_isCompact_of_ne_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsCompact.{u1} α _inst_2 K) (fun (h : IsCompact.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsCompact.{u1} α _inst_2 K) (fun (h : IsCompact.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_topₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- Given a regular measure, any measurable set of finite mass can be approximated from inside by compact sets. -/ @@ -578,6 +722,12 @@ theorem MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [Regular μ] ⦃A : Se Regular.innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩ #align measurable_set.measure_eq_supr_is_compact_of_ne_top MeasurableSet.measure_eq_iSup_isCompact_of_ne_top +/- warning: measure_theory.measure.regular.map -> MeasureTheory.Measure.Regular.map is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u2} β] [_inst_5 : TopologicalSpace.{u2} β] [_inst_6 : T2Space.{u2} β _inst_5] [_inst_7 : BorelSpace.{u2} β _inst_5 _inst_4] [_inst_8 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] (f : Homeomorph.{u1, u2} α β _inst_2 _inst_5), MeasureTheory.Measure.Regular.{u2} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u1, u2} α β _inst_4 _inst_1 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Homeomorph.{u1, u2} α β _inst_2 _inst_5) (fun (_x : Homeomorph.{u1, u2} α β _inst_2 _inst_5) => α -> β) (Homeomorph.hasCoeToFun.{u1, u2} α β _inst_2 _inst_5) f) μ) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : MeasurableSpace.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] {μ : MeasureTheory.Measure.{u2} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u2} α _inst_2 _inst_1] [_inst_4 : MeasurableSpace.{u1} β] [_inst_5 : TopologicalSpace.{u1} β] [_inst_6 : T2Space.{u1} β _inst_5] [_inst_7 : BorelSpace.{u1} β _inst_5 _inst_4] [_inst_8 : MeasureTheory.Measure.Regular.{u2} α _inst_1 _inst_2 μ] (f : Homeomorph.{u2, u1} α β _inst_2 _inst_5), MeasureTheory.Measure.Regular.{u1} β _inst_4 _inst_5 (MeasureTheory.Measure.map.{u2, u1} α β _inst_4 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α (fun (_x : α) => β) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} α β _inst_2 _inst_5) α β (Homeomorph.instEquivLikeHomeomorph.{u2, u1} α β _inst_2 _inst_5))) f) μ) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.mapₓ'. -/ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [T2Space β] [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).regular := by @@ -589,6 +739,12 @@ protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [Topologica (fun K hK => hK.MeasurableSet) fun U hU => hU.MeasurableSet⟩ #align measure_theory.measure.regular.map MeasureTheory.Measure.Regular.map +/- warning: measure_theory.measure.regular.smul -> MeasureTheory.Measure.Regular.smul is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 (SMul.smul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (SMulZeroClass.toHasSmul.{0, 0} ENNReal ENNReal (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (SMulWithZero.toSmulZeroClass.{0, 0} ENNReal ENNReal (MulZeroClass.toHasZero.{0} ENNReal (MulZeroOneClass.toMulZeroClass.{0} ENNReal (MonoidWithZero.toMulZeroOneClass.{0} ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (MulActionWithZero.toSMulWithZero.{0, 0} ENNReal ENNReal (Semiring.toMonoidWithZero.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (AddZeroClass.toHasZero.{0} ENNReal (AddMonoid.toAddZeroClass.{0} ENNReal (AddCommMonoid.toAddMonoid.{0} ENNReal (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))))) (Module.toMulActionWithZero.{0, 0} ENNReal ENNReal (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))) (Algebra.toModule.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))))))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring))) _inst_1) x μ)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 μ] {x : ENNReal}, (Ne.{1} ENNReal x (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.Regular.{u1} α _inst_1 _inst_2 (HSMul.hSMul.{0, u1, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.{u1} α _inst_1) (instHSMul.{0, u1} ENNReal (MeasureTheory.Measure.{u1} α _inst_1) (MeasureTheory.Measure.instSMul.{u1, 0} α ENNReal (Algebra.toSMul.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) (IsScalarTower.right.{0, 0} ENNReal ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal) (CommSemiring.toSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)) (Algebra.id.{0} ENNReal (CanonicallyOrderedCommSemiring.toCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal))) _inst_1)) x μ)) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smulₓ'. -/ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).regular := by haveI := outer_regular.smul μ hx @@ -596,6 +752,7 @@ protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • exact ⟨regular.inner_regular.smul x⟩ #align measure_theory.measure.regular.smul MeasureTheory.Measure.Regular.smul +#print MeasureTheory.Measure.Regular.sigmaFinite /- -- see Note [lower instance priority] /-- A regular measure in a σ-compact space is σ-finite. -/ instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : SigmaFinite μ := @@ -604,11 +761,18 @@ instance (priority := 100) sigmaFinite [SigmaCompactSpace α] [Regular μ] : Sig Finite := fun n => (isCompact_compactCovering α n).measure_lt_top spanning := iUnion_compactCovering α }⟩⟩ #align measure_theory.measure.regular.sigma_finite MeasureTheory.Measure.Regular.sigmaFinite +-/ end Regular namespace WeaklyRegular +/- warning: is_open.exists_lt_is_closed -> IsOpen.exists_lt_isClosed is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) => And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ F)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U)) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) (And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) F)))))) +Case conversion may be inaccurate. Consider using '#align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosedₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/ /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞} @@ -616,6 +780,12 @@ theorem IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOp WeaklyRegular.innerRegular hU r hr #align is_open.exists_lt_is_closed IsOpen.exists_lt_isClosed +/- warning: is_open.measure_eq_supr_is_closed -> IsOpen.measure_eq_iSup_isClosed is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (F : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsClosed.{u1} α _inst_2 F) (fun (h : IsClosed.{u1} α _inst_2 F) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ F))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {{U : Set.{u1} α}}, (IsOpen.{u1} α _inst_2 U) -> (forall (μ : MeasureTheory.Measure.{u1} α _inst_1) [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) U) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (F : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F U) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsClosed.{u1} α _inst_2 F) (fun (h : IsClosed.{u1} α _inst_2 F) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) F))))) +Case conversion may be inaccurate. Consider using '#align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosedₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » U) -/ /-- If `μ` is a weakly regular measure, then any open set can be approximated by a closed subset. -/ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α) @@ -623,12 +793,24 @@ theorem IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : M WeaklyRegular.innerRegular.measure_eq_iSup hU #align is_open.measure_eq_supr_is_closed IsOpen.measure_eq_iSup_isClosed +/- warning: measure_theory.measure.weakly_regular.inner_regular_measurable -> MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsClosed.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ], MeasureTheory.Measure.InnerRegular.{u1} α _inst_1 μ (IsClosed.{u1} α _inst_2) (fun (s : Set.{u1} α) => And (MeasurableSet.{u1} α _inst_1 s) (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurableₓ'. -/ theorem innerRegular_measurable [WeaklyRegular μ] : InnerRegular μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ := WeaklyRegular.innerRegular.measurableSet_of_open isClosed_empty fun _ _ h₁ h₂ => h₁.inter h₂.isClosed_compl #align measure_theory.measure.weakly_regular.inner_regular_measurable MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable +/- warning: measurable_set.exists_is_closed_lt_add -> MeasurableSet.exists_isClosed_lt_add is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K s) => And (IsClosed.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ s) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toHasAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K) ε)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {s : Set.{u1} α}, (MeasurableSet.{u1} α _inst_1 s) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K s) (And (IsClosed.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) s) (HAdd.hAdd.{0, 0, 0} ENNReal ENNReal ENNReal (instHAdd.{0} ENNReal (Distrib.toAdd.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (OrderedSemiring.toSemiring.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.instCanonicallyOrderedCommSemiringENNReal)))))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K) ε)))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_addₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » s) -/ /-- If `s` is a measurable set, a weakly regular measure `μ` is finite on `s`, and `ε` is a positive number, then there exist a closed set `K ⊆ s` such that `μ s < μ K + ε`. -/ @@ -637,6 +819,12 @@ theorem MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α} (hs innerRegular_measurable.exists_subset_lt_add isClosed_empty ⟨hs, hμs⟩ hμs hε #align measurable_set.exists_is_closed_lt_add MeasurableSet.exists_isClosed_lt_add +/- warning: measurable_set.exists_is_closed_diff_lt -> MeasurableSet.exists_isClosed_diff_lt is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) F A) => And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) A F)) ε))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : OpensMeasurableSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {ε : ENNReal}, (Ne.{1} ENNReal ε (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero))) -> (Exists.{succ u1} (Set.{u1} α) (fun (F : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) F A) (And (IsClosed.{u1} α _inst_2 F) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) A F)) ε))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_ltₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (F «expr ⊆ » A) -/ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : @@ -649,6 +837,12 @@ theorem MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyR hF⟩ #align measurable_set.exists_is_closed_diff_lt MeasurableSet.exists_isClosed_diff_lt +/- warning: measurable_set.exists_lt_is_closed_of_ne_top -> MeasurableSet.exists_lt_isClosed_of_ne_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => And (IsClosed.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toHasLt.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) r (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K)))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (forall {r : ENNReal}, (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A)) -> (Exists.{succ u1} (Set.{u1} α) (fun (K : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (And (IsClosed.{u1} α _inst_2 K) (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) r (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K)))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_topₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from inside by closed sets. -/ @@ -658,6 +852,12 @@ theorem MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set innerRegular_measurable ⟨hA, h'A⟩ _ hr #align measurable_set.exists_lt_is_closed_of_ne_top MeasurableSet.exists_lt_isClosed_of_ne_top +/- warning: measurable_set.measure_eq_supr_is_closed_of_ne_top -> MeasurableSet.measure_eq_iSup_isClosed_of_ne_top is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (Eq.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toHasSup.{0} ENNReal (CompleteLattice.toConditionallyCompleteLattice.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))) (IsClosed.{u1} α _inst_2 K) (fun (h : IsClosed.{u1} α _inst_2 K) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ K))))) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] {{A : Set.{u1} α}}, (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (iSup.{0, succ u1} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (Set.{u1} α) (fun (K : Set.{u1} α) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) K A) => iSup.{0, 0} ENNReal (ConditionallyCompleteLattice.toSupSet.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) (IsClosed.{u1} α _inst_2 K) (fun (h : IsClosed.{u1} α _inst_2 K) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) K))))) +Case conversion may be inaccurate. Consider using '#align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_topₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (K «expr ⊆ » A) -/ /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from inside by closed sets. -/ @@ -666,6 +866,12 @@ theorem MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A innerRegular_measurable.measure_eq_iSup ⟨hA, h'A⟩ #align measurable_set.measure_eq_supr_is_closed_of_ne_top MeasurableSet.measure_eq_iSup_isClosed_of_ne_top +/- warning: measure_theory.measure.weakly_regular.restrict_of_measurable_set -> MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : BorelSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α _inst_1) (fun (_x : MeasureTheory.Measure.{u1} α _inst_1) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α _inst_1) μ A) (Top.top.{0} ENNReal (CompleteLattice.toHasTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder)))) -> (MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 (MeasureTheory.Measure.restrict.{u1} α _inst_1 μ A)) +but is expected to have type + forall {α : Type.{u1}} [_inst_1 : MeasurableSpace.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] {μ : MeasureTheory.Measure.{u1} α _inst_1} [_inst_3 : BorelSpace.{u1} α _inst_2 _inst_1] [_inst_4 : MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 μ] (A : Set.{u1} α), (MeasurableSet.{u1} α _inst_1 A) -> (Ne.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α _inst_1 μ) A) (Top.top.{0} ENNReal (CompleteLattice.toTop.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal)))) -> (MeasureTheory.Measure.WeaklyRegular.{u1} α _inst_1 _inst_2 (MeasureTheory.Measure.restrict.{u1} α _inst_1 μ A)) +Case conversion may be inaccurate. Consider using '#align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSetₓ'. -/ /-- The restriction of a weakly regular measure to a measurable set of finite measure is weakly regular. -/ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α) @@ -682,6 +888,7 @@ theorem restrict_of_measurableSet [BorelSpace α] [WeaklyRegular μ] (A : Set α rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)] #align measure_theory.measure.weakly_regular.restrict_of_measurable_set MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet +#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure /- -- see Note [lower instance priority] /-- Any finite measure on a metric space (or even a pseudo emetric space) is weakly regular. -/ instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _} @@ -689,11 +896,13 @@ instance (priority := 100) of_pseudoEMetricSpace_of_finiteMeasure {X : Type _} WeaklyRegular μ := (InnerRegular.of_pseudoEMetricSpace μ).weaklyRegular_of_finite μ #align measure_theory.measure.weakly_regular.of_pseudo_emetric_space_of_is_finite_measure MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetricSpace_of_finiteMeasure +-/ +#print MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite /- -- see Note [lower instance priority] /-- Any locally finite measure on a second countable metric space (or even a pseudo emetric space) is weakly regular. -/ -instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite {X : Type _} +instance (priority := 100) of_pseudoEMetric_secondCountable_of_locallyFinite {X : Type _} [PseudoEMetricSpace X] [TopologicalSpace.SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [LocallyFiniteMeasure μ] : WeaklyRegular μ := haveI : outer_regular μ := @@ -702,12 +911,14 @@ instance (priority := 100) of_pseudo_emetric_second_countable_of_locally_finite have : Fact (μ U < ∞) := ⟨hU.2⟩ exact ⟨hU.1, inferInstance⟩ ⟨inner_regular.of_pseudo_emetric_space μ⟩ -#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudo_emetric_second_countable_of_locally_finite +#align measure_theory.measure.weakly_regular.of_pseudo_emetric_second_countable_of_locally_finite MeasureTheory.Measure.WeaklyRegular.of_pseudoEMetric_secondCountable_of_locallyFinite +-/ end WeaklyRegular attribute [local instance] EMetric.secondCountable_of_sigmaCompact +#print MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure /- -- see Note [lower instance priority] /-- Any locally finite measure on a `σ`-compact (e)metric space is regular. -/ instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure {X : Type _} @@ -717,6 +928,7 @@ instance (priority := 100) Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure lt_top_of_isCompact K hK := hK.measure_lt_top InnerRegular := (InnerRegular.isCompact_isClosed μ).trans (InnerRegular.of_pseudoEMetricSpace μ) #align measure_theory.measure.regular.of_sigma_compact_space_of_is_locally_finite_measure MeasureTheory.Measure.Regular.of_sigmaCompactSpace_of_locallyFiniteMeasure +-/ end Measure diff --git a/Mathbin/MeasureTheory/Measure/Stieltjes.lean b/Mathbin/MeasureTheory/Measure/Stieltjes.lean index 2c84fe8c54..6ff5868c76 100644 --- a/Mathbin/MeasureTheory/Measure/Stieltjes.lean +++ b/Mathbin/MeasureTheory/Measure/Stieltjes.lean @@ -35,6 +35,12 @@ open Filter Set open Topology +/- warning: infi_Ioi_eq_infi_rat_gt -> iInf_Ioi_eq_iInf_rat_gt is a dubious translation: +lean 3 declaration is + forall {f : Real -> Real} (x : Real), (BddBelow.{0} Real Real.preorder (Set.image.{0, 0} Real Real f (Set.Ioi.{0} Real Real.preorder x))) -> (Monotone.{0, 0} Real Real Real.preorder Real.preorder f) -> (Eq.{1} Real (iInf.{0, 1} Real Real.hasInf (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) (fun (r : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) => f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (coeSubtype.{1} Real (fun (x_1 : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x_1 (Set.Ioi.{0} Real Real.preorder x)))))) r))) (iInf.{0, 1} Real Real.hasInf (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) (fun (q : Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) => f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) Real (HasLiftT.mk.{1, 1} (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) Real (CoeTCₓ.coe.{1, 1} (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) Real (coeTrans.{1, 1, 1} (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q'))) Rat Real (Rat.castCoe.{0} Real Real.hasRatCast) (coeSubtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) q')))))) q)))) +but is expected to have type + forall {f : Real -> Real} (x : Real), (BddBelow.{0} Real Real.instPreorderReal (Set.image.{0, 0} Real Real f (Set.Ioi.{0} Real Real.instPreorderReal x))) -> (Monotone.{0, 0} Real Real Real.instPreorderReal Real.instPreorderReal f) -> (Eq.{1} Real (iInf.{0, 1} Real Real.instInfSetReal (Set.Elem.{0} Real (Set.Ioi.{0} Real Real.instPreorderReal x)) (fun (r : Set.Elem.{0} Real (Set.Ioi.{0} Real Real.instPreorderReal x)) => f (Subtype.val.{1} Real (fun (x_1 : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x_1 (Set.Ioi.{0} Real Real.instPreorderReal x)) r))) (iInf.{0, 1} Real Real.instInfSetReal (Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast q'))) (fun (q : Subtype.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast q'))) => f (Rat.cast.{0} Real Real.ratCast (Subtype.val.{1} Rat (fun (q' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast q')) q))))) +Case conversion may be inaccurate. Consider using '#align infi_Ioi_eq_infi_rat_gt iInf_Ioi_eq_iInf_rat_gtₓ'. -/ theorem iInf_Ioi_eq_iInf_rat_gt {f : ℝ → ℝ} (x : ℝ) (hf : BddBelow (f '' Ioi x)) (hf_mono : Monotone f) : (⨅ r : Ioi x, f r) = ⨅ q : { q' : ℚ // x < q' }, f q := by @@ -62,6 +68,12 @@ theorem iInf_Ioi_eq_iInf_rat_gt {f : ℝ → ℝ} (x : ℝ) (hf : BddBelow (f '' norm_cast #align infi_Ioi_eq_infi_rat_gt iInf_Ioi_eq_iInf_rat_gt +/- warning: right_lim_eq_of_tendsto -> rightLim_eq_of_tendsto is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [hα : TopologicalSpace.{u1} α] [h'α : OrderTopology.{u1} α hα (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1))))] [_inst_3 : T2Space.{u2} β _inst_2] {f : α -> β} {a : α} {y : β}, (Ne.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α hα a (Set.Ioi.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))) -> (Filter.Tendsto.{u1, u2} α β f (nhdsWithin.{u1} α hα a (Set.Ioi.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) (nhds.{u2} β _inst_2 y)) -> (Eq.{succ u2} β (Function.rightLim.{u1, u2} α β _inst_1 _inst_2 f a) y) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] [hα : TopologicalSpace.{u2} α] [h'α : OrderTopology.{u2} α hα (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1)))))] [_inst_3 : T2Space.{u1} β _inst_2] {f : α -> β} {a : α} {y : β}, (Ne.{succ u2} (Filter.{u2} α) (nhdsWithin.{u2} α hα a (Set.Ioi.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) (Bot.bot.{u2} (Filter.{u2} α) (CompleteLattice.toBot.{u2} (Filter.{u2} α) (Filter.instCompleteLatticeFilter.{u2} α)))) -> (Filter.Tendsto.{u2, u1} α β f (nhdsWithin.{u2} α hα a (Set.Ioi.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) (nhds.{u1} β _inst_2 y)) -> (Eq.{succ u1} β (Function.rightLim.{u2, u1} α β _inst_1 _inst_2 f a) y) +Case conversion may be inaccurate. Consider using '#align right_lim_eq_of_tendsto rightLim_eq_of_tendstoₓ'. -/ -- todo after the port: move to topology/algebra/order/left_right_lim theorem rightLim_eq_of_tendsto {α β : Type _} [LinearOrder α] [TopologicalSpace β] [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} @@ -69,6 +81,12 @@ theorem rightLim_eq_of_tendsto {α β : Type _} [LinearOrder α] [TopologicalSpa @leftLim_eq_of_tendsto αᵒᵈ _ _ _ _ _ _ f a y h h' #align right_lim_eq_of_tendsto rightLim_eq_of_tendsto +/- warning: right_lim_eq_Inf -> rightLim_eq_sInf is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_4 : OrderTopology.{u2} β _inst_2 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_3)))))] {f : α -> β}, (Monotone.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_3))))) f) -> (forall {x : α} [_inst_5 : TopologicalSpace.{u1} α] [_inst_6 : OrderTopology.{u1} α _inst_5 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1))))], (Ne.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_5 x (Set.Ioi.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) x)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))) -> (Eq.{succ u2} β (Function.rightLim.{u1, u2} α β _inst_1 _inst_2 f x) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_3)) (Set.image.{u1, u2} α β f (Set.Ioi.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) x))))) +but is expected to have type + forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_4 : OrderTopology.{u1} β _inst_2 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_3)))))] {f : α -> β}, (Monotone.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_3))))) f) -> (forall {x : α} [_inst_5 : TopologicalSpace.{u2} α] [_inst_6 : OrderTopology.{u2} α _inst_5 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1)))))], (Ne.{succ u2} (Filter.{u2} α) (nhdsWithin.{u2} α _inst_5 x (Set.Ioi.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) x)) (Bot.bot.{u2} (Filter.{u2} α) (CompleteLattice.toBot.{u2} (Filter.{u2} α) (Filter.instCompleteLatticeFilter.{u2} α)))) -> (Eq.{succ u1} β (Function.rightLim.{u2, u1} α β _inst_1 _inst_2 f x) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_3)) (Set.image.{u2, u1} α β f (Set.Ioi.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) x))))) +Case conversion may be inaccurate. Consider using '#align right_lim_eq_Inf rightLim_eq_sInfₓ'. -/ -- todo after the port: move to topology/algebra/order/left_right_lim theorem rightLim_eq_sInf {α β : Type _} [LinearOrder α] [TopologicalSpace β] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} @@ -77,6 +95,7 @@ theorem rightLim_eq_sInf {α β : Type _} [LinearOrder α] [TopologicalSpace β] rightLim_eq_of_tendsto h (hf.tendsto_nhdsWithin_Ioi x) #align right_lim_eq_Inf rightLim_eq_sInf +#print exists_seq_monotone_tendsto_atTop_atTop /- -- todo after the port: move to order/filter/at_top_bot theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type _) [SemilatticeSup α] [Nonempty α] [(atTop : Filter α).IsCountablyGenerated] : @@ -106,13 +125,22 @@ theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type _) [SemilatticeSup α refine' Finset.le_sup'_of_le _ _ le_rfl rw [Finset.mem_range_succ_iff] #align exists_seq_monotone_tendsto_at_top_at_top exists_seq_monotone_tendsto_atTop_atTop +-/ +#print exists_seq_antitone_tendsto_atTop_atBot /- theorem exists_seq_antitone_tendsto_atTop_atBot (α : Type _) [SemilatticeInf α] [Nonempty α] [h2 : (atBot : Filter α).IsCountablyGenerated] : ∃ xs : ℕ → α, Antitone xs ∧ Tendsto xs atTop atBot := @exists_seq_monotone_tendsto_atTop_atTop αᵒᵈ _ _ h2 #align exists_seq_antitone_tendsto_at_top_at_bot exists_seq_antitone_tendsto_atTop_atBot +-/ +/- warning: supr_eq_supr_subseq_of_antitone -> iSup_eq_iSup_subseq_of_antitone is a dubious translation: +lean 3 declaration is + forall {ι₁ : Type.{u1}} {ι₂ : Type.{u2}} {α : Type.{u3}} [_inst_1 : Preorder.{u2} ι₂] [_inst_2 : CompleteLattice.{u3} α] {l : Filter.{u1} ι₁} [_inst_3 : Filter.NeBot.{u1} ι₁ l] {f : ι₂ -> α} {φ : ι₁ -> ι₂}, (Antitone.{u2, u3} ι₂ α _inst_1 (PartialOrder.toPreorder.{u3} α (CompleteSemilatticeInf.toPartialOrder.{u3} α (CompleteLattice.toCompleteSemilatticeInf.{u3} α _inst_2))) f) -> (Filter.Tendsto.{u1, u2} ι₁ ι₂ φ l (Filter.atBot.{u2} ι₂ _inst_1)) -> (Eq.{succ u3} α (iSup.{u3, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u3} α (CompleteLattice.toConditionallyCompleteLattice.{u3} α _inst_2)) ι₂ (fun (i : ι₂) => f i)) (iSup.{u3, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u3} α (CompleteLattice.toConditionallyCompleteLattice.{u3} α _inst_2)) ι₁ (fun (i : ι₁) => f (φ i)))) +but is expected to have type + forall {ι₁ : Type.{u3}} {ι₂ : Type.{u2}} {α : Type.{u1}} [_inst_1 : Preorder.{u2} ι₂] [_inst_2 : CompleteLattice.{u1} α] {l : Filter.{u3} ι₁} [_inst_3 : Filter.NeBot.{u3} ι₁ l] {f : ι₂ -> α} {φ : ι₁ -> ι₂}, (Antitone.{u2, u1} ι₂ α _inst_1 (PartialOrder.toPreorder.{u1} α (OmegaCompletePartialOrder.toPartialOrder.{u1} α (CompleteLattice.instOmegaCompletePartialOrder.{u1} α _inst_2))) f) -> (Filter.Tendsto.{u3, u2} ι₁ ι₂ φ l (Filter.atBot.{u2} ι₂ _inst_1)) -> (Eq.{succ u1} α (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι₂ (fun (i : ι₂) => f i)) (iSup.{u1, succ u3} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι₁ (fun (i : ι₁) => f (φ i)))) +Case conversion may be inaccurate. Consider using '#align supr_eq_supr_subseq_of_antitone iSup_eq_iSup_subseq_of_antitoneₓ'. -/ -- todo after the port: move to topology/algebra/order/monotone_convergence theorem iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type _} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.ne_bot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f) @@ -130,6 +158,12 @@ variable {α : Type _} {mα : MeasurableSpace α} include mα +/- warning: measure_theory.tendsto_measure_Ico_at_top -> MeasureTheory.tendsto_measure_Ico_atTop is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : NoMaxOrder.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] [_inst_3 : Filter.IsCountablyGenerated.{u1} α (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα) (a : α), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Ico.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) a x)) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.topologicalSpace (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) a))) +but is expected to have type + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : NoMaxOrder.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] [_inst_3 : Filter.IsCountablyGenerated.{u1} α (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα) (a : α), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Ico.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) a x)) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_measure_Ico_at_top MeasureTheory.tendsto_measure_Ico_atTopₓ'. -/ theorem tendsto_measure_Ico_atTop [SemilatticeSup α] [NoMaxOrder α] [(atTop : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) : Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a))) := @@ -150,6 +184,12 @@ theorem tendsto_measure_Ico_atTop [SemilatticeSup α] [NoMaxOrder α] exact Monotone.directed_le fun i j hij => Ico_subset_Ico_right (hxs_mono hij) #align measure_theory.tendsto_measure_Ico_at_top MeasureTheory.tendsto_measure_Ico_atTop +/- warning: measure_theory.tendsto_measure_Ioc_at_bot -> MeasureTheory.tendsto_measure_Ioc_atBot is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeInf.{u1} α] [_inst_2 : NoMinOrder.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] [_inst_3 : Filter.IsCountablyGenerated.{u1} α (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα) (a : α), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Ioc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) x a)) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.topologicalSpace (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) a))) +but is expected to have type + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeInf.{u1} α] [_inst_2 : NoMinOrder.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] [_inst_3 : Filter.IsCountablyGenerated.{u1} α (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα) (a : α), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Ioc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) x a)) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) a))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_measure_Ioc_at_bot MeasureTheory.tendsto_measure_Ioc_atBotₓ'. -/ theorem tendsto_measure_Ioc_atBot [SemilatticeInf α] [NoMinOrder α] [(atBot : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) : Tendsto (fun x => μ (Ioc x a)) atBot (𝓝 (μ (Iic a))) := @@ -170,6 +210,12 @@ theorem tendsto_measure_Ioc_atBot [SemilatticeInf α] [NoMinOrder α] exact Monotone.directed_le fun i j hij => Ioc_subset_Ioc_left (hxs_mono hij) #align measure_theory.tendsto_measure_Ioc_at_bot MeasureTheory.tendsto_measure_Ioc_atBot +/- warning: measure_theory.tendsto_measure_Iic_at_top -> MeasureTheory.tendsto_measure_Iic_atTop is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : Filter.IsCountablyGenerated.{u1} α (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) x)) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.topologicalSpace (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : Filter.IsCountablyGenerated.{u1} α (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) x)) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_measure_Iic_at_top MeasureTheory.tendsto_measure_Iic_atTopₓ'. -/ theorem tendsto_measure_Iic_atTop [SemilatticeSup α] [(atTop : Filter α).IsCountablyGenerated] (μ : Measure α) : Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ)) := by @@ -191,6 +237,12 @@ theorem tendsto_measure_Iic_atTop [SemilatticeSup α] [(atTop : Filter α).IsCou exact Monotone.directed_le fun i j hij => Iic_subset_Iic.mpr (hxs_mono hij) #align measure_theory.tendsto_measure_Iic_at_top MeasureTheory.tendsto_measure_Iic_atTop +/- warning: measure_theory.tendsto_measure_Ici_at_bot -> MeasureTheory.tendsto_measure_Ici_atBot is a dubious translation: +lean 3 declaration is + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeInf.{u1} α] [h : Filter.IsCountablyGenerated.{u1} α (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) x)) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.topologicalSpace (coeFn.{succ u1, succ u1} (MeasureTheory.Measure.{u1} α mα) (fun (_x : MeasureTheory.Measure.{u1} α mα) => (Set.{u1} α) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{u1} α mα) μ (Set.univ.{u1} α))) +but is expected to have type + forall {α : Type.{u1}} {mα : MeasurableSpace.{u1} α} [_inst_1 : SemilatticeInf.{u1} α] [h : Filter.IsCountablyGenerated.{u1} α (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))] (μ : MeasureTheory.Measure.{u1} α mα), Filter.Tendsto.{u1, 0} α ENNReal (fun (x : α) => MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)) x)) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1))) (nhds.{0} ENNReal ENNReal.instTopologicalSpaceENNReal (MeasureTheory.OuterMeasure.measureOf.{u1} α (MeasureTheory.Measure.toOuterMeasure.{u1} α mα μ) (Set.univ.{u1} α))) +Case conversion may be inaccurate. Consider using '#align measure_theory.tendsto_measure_Ici_at_bot MeasureTheory.tendsto_measure_Ici_atBotₓ'. -/ theorem tendsto_measure_Ici_atBot [SemilatticeInf α] [h : (atBot : Filter α).IsCountablyGenerated] (μ : Measure α) : Tendsto (fun x => μ (Ici x)) atBot (𝓝 (μ univ)) := @tendsto_measure_Iic_atTop αᵒᵈ _ _ h μ @@ -211,12 +263,14 @@ open BigOperators ENNReal NNReal Topology MeasureTheory /-! ### Basic properties of Stieltjes functions -/ +#print StieltjesFunction /- /-- Bundled monotone right-continuous real functions, used to construct Stieltjes measures. -/ structure StieltjesFunction where toFun : ℝ → ℝ mono' : Monotone to_fun right_continuous' : ∀ x, ContinuousWithinAt to_fun (Ici x) x #align stieltjes_function StieltjesFunction +-/ namespace StieltjesFunction @@ -227,20 +281,32 @@ initialize_simps_projections StieltjesFunction (toFun → apply) variable (f : StieltjesFunction) +#print StieltjesFunction.mono /- theorem mono : Monotone f := f.mono' #align stieltjes_function.mono StieltjesFunction.mono +-/ +#print StieltjesFunction.right_continuous /- theorem right_continuous (x : ℝ) : ContinuousWithinAt f (Ici x) x := f.right_continuous' x #align stieltjes_function.right_continuous StieltjesFunction.right_continuous +-/ +#print StieltjesFunction.rightLim_eq /- theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x := by rw [← f.mono.continuous_within_at_Ioi_iff_right_lim_eq, continuousWithinAt_Ioi_iff_Ici] exact f.right_continuous' x #align stieltjes_function.right_lim_eq StieltjesFunction.rightLim_eq +-/ +/- warning: stieltjes_function.infi_Ioi_eq -> StieltjesFunction.iInf_Ioi_eq is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (x : Real), Eq.{1} Real (iInf.{0, 1} Real Real.hasInf (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) (fun (r : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) => coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Ioi.{0} Real Real.preorder x)) Real (coeSubtype.{1} Real (fun (x_1 : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x_1 (Set.Ioi.{0} Real Real.preorder x)))))) r))) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f x) +but is expected to have type + forall (f : StieltjesFunction) (x : Real), Eq.{1} Real (iInf.{0, 1} Real Real.instInfSetReal (Set.Elem.{0} Real (Set.Ioi.{0} Real Real.instPreorderReal x)) (fun (r : Set.Elem.{0} Real (Set.Ioi.{0} Real Real.instPreorderReal x)) => StieltjesFunction.toFun f (Subtype.val.{1} Real (fun (x_1 : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x_1 (Set.Ioi.{0} Real Real.instPreorderReal x)) r))) (StieltjesFunction.toFun f x) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.infi_Ioi_eq StieltjesFunction.iInf_Ioi_eqₓ'. -/ theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : (⨅ r : Ioi x, f r) = f x := by suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.right_lim_eq] @@ -249,6 +315,12 @@ theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : (⨅ r : Ioi x, f r) = f infer_instance #align stieltjes_function.infi_Ioi_eq StieltjesFunction.iInf_Ioi_eq +/- warning: stieltjes_function.infi_rat_gt_eq -> StieltjesFunction.iInf_rat_gt_eq is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (x : Real), Eq.{1} Real (iInf.{0, 1} Real Real.hasInf (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) (fun (r : Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) => coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) Real (HasLiftT.mk.{1, 1} (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) Real (CoeTCₓ.coe.{1, 1} (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) Real (coeTrans.{1, 1, 1} (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r'))) Rat Real (Rat.castCoe.{0} Real Real.hasRatCast) (coeSubtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.hasLt x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Rat Real (HasLiftT.mk.{1, 1} Rat Real (CoeTCₓ.coe.{1, 1} Rat Real (Rat.castCoe.{0} Real Real.hasRatCast))) r')))))) r))) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f x) +but is expected to have type + forall (f : StieltjesFunction) (x : Real), Eq.{1} Real (iInf.{0, 1} Real Real.instInfSetReal (Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast r'))) (fun (r : Subtype.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast r'))) => StieltjesFunction.toFun f (Rat.cast.{0} Real Real.ratCast (Subtype.val.{1} Rat (fun (r' : Rat) => LT.lt.{0} Real Real.instLTReal x (Rat.cast.{0} Real Real.ratCast r')) r)))) (StieltjesFunction.toFun f x) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.infi_rat_gt_eq StieltjesFunction.iInf_rat_gt_eqₓ'. -/ theorem iInf_rat_gt_eq (f : StieltjesFunction) (x : ℝ) : (⨅ r : { r' : ℚ // x < r' }, f r) = f x := by rw [← infi_Ioi_eq f x] @@ -258,6 +330,7 @@ theorem iInf_rat_gt_eq (f : StieltjesFunction) (x : ℝ) : (⨅ r : { r' : ℚ / exact f.mono (le_of_lt hy_mem) #align stieltjes_function.infi_rat_gt_eq StieltjesFunction.iInf_rat_gt_eq +#print StieltjesFunction.id /- /-- The identity of `ℝ` as a Stieltjes function, used to construct Lebesgue measure. -/ @[simps] protected def id : StieltjesFunction where @@ -265,16 +338,20 @@ protected def id : StieltjesFunction where mono' x y := id right_continuous' x := continuousWithinAt_id #align stieltjes_function.id StieltjesFunction.id +-/ +#print StieltjesFunction.id_leftLim /- @[simp] theorem id_leftLim (x : ℝ) : leftLim StieltjesFunction.id x = x := tendsto_nhds_unique (StieltjesFunction.id.mono.tendsto_leftLim x) <| continuousAt_id.Tendsto.mono_left nhdsWithin_le_nhds #align stieltjes_function.id_left_lim StieltjesFunction.id_leftLim +-/ instance : Inhabited StieltjesFunction := ⟨StieltjesFunction.id⟩ +#print Monotone.stieltjesFunction /- /-- If a function `f : ℝ → ℝ` is monotone, then the function mapping `x` to the right limit of `f` at `x` is a Stieltjes function, i.e., it is monotone and right-continuous. -/ noncomputable def Monotone.stieltjesFunction {f : ℝ → ℝ} (hf : Monotone f) : StieltjesFunction @@ -297,12 +374,16 @@ noncomputable def Monotone.stieltjesFunction {f : ℝ → ℝ} (hf : Monotone f) _ < u := (h'y ⟨hz.1.trans_lt za, ay.le⟩).2 #align monotone.stieltjes_function Monotone.stieltjesFunction +-/ +#print Monotone.stieltjesFunction_eq /- theorem Monotone.stieltjesFunction_eq {f : ℝ → ℝ} (hf : Monotone f) (x : ℝ) : hf.StieltjesFunction x = rightLim f x := rfl #align monotone.stieltjes_function_eq Monotone.stieltjesFunction_eq +-/ +#print StieltjesFunction.countable_leftLim_ne /- theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftLim f x ≠ f x } := by apply countable.mono _ f.mono.countable_not_continuous_at @@ -310,22 +391,37 @@ theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftL apply hx exact tendsto_nhds_unique (f.mono.tendsto_left_lim x) (h'x.tendsto.mono_left nhdsWithin_le_nhds) #align stieltjes_function.countable_left_lim_ne StieltjesFunction.countable_leftLim_ne +-/ /-! ### The outer measure associated to a Stieltjes function -/ /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (a b) -/ +#print StieltjesFunction.length /- /-- Length of an interval. This is the largest monotone function which correctly measures all intervals. -/ def length (s : Set ℝ) : ℝ≥0∞ := ⨅ (a) (b) (h : s ⊆ Ioc a b), ofReal (f b - f a) #align stieltjes_function.length StieltjesFunction.length +-/ +/- warning: stieltjes_function.length_empty -> StieltjesFunction.length_empty is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction), Eq.{1} ENNReal (StieltjesFunction.length f (EmptyCollection.emptyCollection.{0} (Set.{0} Real) (Set.hasEmptyc.{0} Real))) (OfNat.ofNat.{0} ENNReal 0 (OfNat.mk.{0} ENNReal 0 (Zero.zero.{0} ENNReal ENNReal.hasZero))) +but is expected to have type + forall (f : StieltjesFunction), Eq.{1} ENNReal (StieltjesFunction.length f (EmptyCollection.emptyCollection.{0} (Set.{0} Real) (Set.instEmptyCollectionSet.{0} Real))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instENNRealZero)) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.length_empty StieltjesFunction.length_emptyₓ'. -/ @[simp] theorem length_empty : f.length ∅ = 0 := nonpos_iff_eq_zero.1 <| iInf_le_of_le 0 <| iInf_le_of_le 0 <| by simp #align stieltjes_function.length_empty StieltjesFunction.length_empty +/- warning: stieltjes_function.length_Ioc -> StieltjesFunction.length_Ioc is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (StieltjesFunction.length f (Set.Ioc.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f b) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (StieltjesFunction.length f (Set.Ioc.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f b) (StieltjesFunction.toFun f a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.length_Ioc StieltjesFunction.length_Iocₓ'. -/ @[simp] theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) := by @@ -339,21 +435,41 @@ theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) := exact Real.toNNReal_le_toNNReal (sub_le_sub (f.mono h₁) (f.mono h₂)) #align stieltjes_function.length_Ioc StieltjesFunction.length_Ioc +/- warning: stieltjes_function.length_mono -> StieltjesFunction.length_mono is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {s₁ : Set.{0} Real} {s₂ : Set.{0} Real}, (HasSubset.Subset.{0} (Set.{0} Real) (Set.hasSubset.{0} Real) s₁ s₂) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (StieltjesFunction.length f s₁) (StieltjesFunction.length f s₂)) +but is expected to have type + forall (f : StieltjesFunction) {s₁ : Set.{0} Real} {s₂ : Set.{0} Real}, (HasSubset.Subset.{0} (Set.{0} Real) (Set.instHasSubsetSet.{0} Real) s₁ s₂) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (StieltjesFunction.length f s₁) (StieltjesFunction.length f s₂)) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.length_mono StieltjesFunction.length_monoₓ'. -/ theorem length_mono {s₁ s₂ : Set ℝ} (h : s₁ ⊆ s₂) : f.length s₁ ≤ f.length s₂ := iInf_mono fun a => biInf_mono fun b => h.trans #align stieltjes_function.length_mono StieltjesFunction.length_mono open MeasureTheory +#print StieltjesFunction.outer /- /-- The Stieltjes outer measure associated to a Stieltjes function. -/ protected def outer : OuterMeasure ℝ := OuterMeasure.ofFunction f.length f.length_empty #align stieltjes_function.outer StieltjesFunction.outer +-/ +/- warning: stieltjes_function.outer_le_length -> StieltjesFunction.outer_le_length is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (s : Set.{0} Real), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (coeFn.{1, 1} (MeasureTheory.OuterMeasure.{0} Real) (fun (_x : MeasureTheory.OuterMeasure.{0} Real) => (Set.{0} Real) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{0} Real) (StieltjesFunction.outer f) s) (StieltjesFunction.length f s) +but is expected to have type + forall (f : StieltjesFunction) (s : Set.{0} Real), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (MeasureTheory.OuterMeasure.measureOf.{0} Real (StieltjesFunction.outer f) s) (StieltjesFunction.length f s) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.outer_le_length StieltjesFunction.outer_le_lengthₓ'. -/ theorem outer_le_length (s : Set ℝ) : f.outer s ≤ f.length s := OuterMeasure.ofFunction_le _ #align stieltjes_function.outer_le_length StieltjesFunction.outer_le_length +/- warning: stieltjes_function.length_subadditive_Icc_Ioo -> StieltjesFunction.length_subadditive_Icc_Ioo is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {a : Real} {b : Real} {c : Nat -> Real} {d : Nat -> Real}, (HasSubset.Subset.{0} (Set.{0} Real) (Set.hasSubset.{0} Real) (Set.Icc.{0} Real Real.preorder a b) (Set.iUnion.{0, 1} Real Nat (fun (i : Nat) => Set.Ioo.{0} Real Real.preorder (c i) (d i)))) -> (LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f b) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a))) (tsum.{0, 0} ENNReal (OrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (OrderedSemiring.toOrderedAddCommMonoid.{0} ENNReal (OrderedCommSemiring.toOrderedSemiring.{0} ENNReal (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENNReal ENNReal.canonicallyOrderedCommSemiring)))) ENNReal.topologicalSpace Nat (fun (i : Nat) => ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f (d i)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f (c i)))))) +but is expected to have type + forall (f : StieltjesFunction) {a : Real} {b : Real} {c : Nat -> Real} {d : Nat -> Real}, (HasSubset.Subset.{0} (Set.{0} Real) (Set.instHasSubsetSet.{0} Real) (Set.Icc.{0} Real Real.instPreorderReal a b) (Set.iUnion.{0, 1} Real Nat (fun (i : Nat) => Set.Ioo.{0} Real Real.instPreorderReal (c i) (d i)))) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (OmegaCompletePartialOrder.toPartialOrder.{0} ENNReal (CompleteLattice.instOmegaCompletePartialOrder.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f b) (StieltjesFunction.toFun f a))) (tsum.{0, 0} ENNReal (LinearOrderedAddCommMonoid.toAddCommMonoid.{0} ENNReal (LinearOrderedAddCommMonoidWithTop.toLinearOrderedAddCommMonoid.{0} ENNReal ENNReal.instLinearOrderedAddCommMonoidWithTopENNReal)) ENNReal.instTopologicalSpaceENNReal Nat (fun (i : Nat) => ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f (d i)) (StieltjesFunction.toFun f (c i)))))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.length_subadditive_Icc_Ioo StieltjesFunction.length_subadditive_Icc_Iooₓ'. -/ /-- If a compact interval `[a, b]` is covered by a union of open interval `(c i, d i)`, then `f b - f a ≤ ∑ f (d i) - f (c i)`. This is an auxiliary technical statement to prove the same statement for half-open intervals, the point of the current statement being that one can use @@ -393,6 +509,12 @@ theorem length_subadditive_Icc_Ioo {a b : ℝ} {c d : ℕ → ℝ} (ss : Icc a b refine' (cv ⟨h₁, le_trans h₂ (le_of_lt cb)⟩).resolve_left (mt And.left (not_lt_of_le h₂)) #align stieltjes_function.length_subadditive_Icc_Ioo StieltjesFunction.length_subadditive_Icc_Ioo +/- warning: stieltjes_function.outer_Ioc -> StieltjesFunction.outer_Ioc is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.OuterMeasure.{0} Real) (fun (_x : MeasureTheory.OuterMeasure.{0} Real) => (Set.{0} Real) -> ENNReal) (MeasureTheory.OuterMeasure.instCoeFun.{0} Real) (StieltjesFunction.outer f) (Set.Ioc.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f b) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (StieltjesFunction.outer f) (Set.Ioc.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f b) (StieltjesFunction.toFun f a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.outer_Ioc StieltjesFunction.outer_Iocₓ'. -/ @[simp] theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) := by @@ -464,6 +586,7 @@ theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) := #align stieltjes_function.outer_Ioc StieltjesFunction.outer_Ioc +#print StieltjesFunction.measurableSet_Ioi /- theorem measurableSet_Ioi {c : ℝ} : measurable_set[f.outer.caratheodory] (Ioi c) := by apply outer_measure.of_function_caratheodory fun t => _ @@ -488,7 +611,9 @@ theorem measurableSet_Ioi {c : ℝ} : measurable_set[f.outer.caratheodory] (Ioi simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right, sup_eq_max, le_refl, Ioc_eq_empty, add_zero, max_eq_left, f.length_empty, not_lt] #align stieltjes_function.measurable_set_Ioi StieltjesFunction.measurableSet_Ioi +-/ +#print StieltjesFunction.outer_trim /- theorem outer_trim : f.outer.trim = f.outer := by refine' le_antisymm (fun s => _) (outer_measure.le_trim _) @@ -516,17 +641,21 @@ theorem outer_trim : f.outer.trim = f.outer := apply iInf_le_of_le (MeasurableSet.iUnion fun i => (hg i).2.1) _ exact le_trans (f.outer.Union _) (ENNReal.tsum_le_tsum fun i => (hg i).2.2) #align stieltjes_function.outer_trim StieltjesFunction.outer_trim +-/ +#print StieltjesFunction.borel_le_measurable /- theorem borel_le_measurable : borel ℝ ≤ f.outer.caratheodory := by rw [borel_eq_generateFrom_Ioi] refine' MeasurableSpace.generateFrom_le _ simp (config := { contextual := true }) [f.measurable_set_Ioi] #align stieltjes_function.borel_le_measurable StieltjesFunction.borel_le_measurable +-/ /-! ### The measure associated to a Stieltjes function -/ +#print StieltjesFunction.measure /- /-- The measure associated to a Stieltjes function, giving mass `f b - f a` to the interval `(a, b]`. -/ protected irreducible_def measure : Measure ℝ := @@ -535,7 +664,14 @@ protected irreducible_def measure : Measure ℝ := f.outer.iUnion_eq_of_caratheodory fun i => f.borel_le_measurable _ (hs i) trimmed := f.outer_trim } #align stieltjes_function.measure StieltjesFunction.measure +-/ +/- warning: stieltjes_function.measure_Ioc -> StieltjesFunction.measure_Ioc is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Ioc.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f b) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Ioc.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f b) (StieltjesFunction.toFun f a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Ioc StieltjesFunction.measure_Iocₓ'. -/ @[simp] theorem measure_Ioc (a b : ℝ) : f.Measure (Ioc a b) = ofReal (f b - f a) := by @@ -543,6 +679,12 @@ theorem measure_Ioc (a b : ℝ) : f.Measure (Ioc a b) = ofReal (f b - f a) := exact f.outer_Ioc a b #align stieltjes_function.measure_Ioc StieltjesFunction.measure_Ioc +/- warning: stieltjes_function.measure_singleton -> StieltjesFunction.measure_singleton is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Singleton.singleton.{0, 0} Real (Set.{0} Real) (Set.hasSingleton.{0} Real) a)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Singleton.singleton.{0, 0} Real (Set.{0} Real) (Set.instSingletonSet.{0} Real) a)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f a) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_singleton StieltjesFunction.measure_singletonₓ'. -/ @[simp] theorem measure_singleton (a : ℝ) : f.Measure {a} = ofReal (f a - leftLim f a) := by @@ -574,6 +716,12 @@ theorem measure_singleton (a : ℝ) : f.Measure {a} = ofReal (f a - leftLim f a) exact tendsto_nhds_unique L1 L2 #align stieltjes_function.measure_singleton StieltjesFunction.measure_singleton +/- warning: stieltjes_function.measure_Icc -> StieltjesFunction.measure_Icc is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Icc.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f b) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Icc.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f b) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Icc StieltjesFunction.measure_Iccₓ'. -/ @[simp] theorem measure_Icc (a b : ℝ) : f.Measure (Icc a b) = ofReal (f b - leftLim f a) := by @@ -586,6 +734,12 @@ theorem measure_Icc (a b : ℝ) : f.Measure (Icc a b) = ofReal (f b - leftLim f simp [ENNReal.ofReal_eq_zero, f.mono.le_left_lim hab] #align stieltjes_function.measure_Icc StieltjesFunction.measure_Icc +/- warning: stieltjes_function.measure_Ioo -> StieltjesFunction.measure_Ioo is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {a : Real} {b : Real}, Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Ioo.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) b) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f a))) +but is expected to have type + forall (f : StieltjesFunction) {a : Real} {b : Real}, Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Ioo.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) b) (StieltjesFunction.toFun f a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Ioo StieltjesFunction.measure_Iooₓ'. -/ @[simp] theorem measure_Ioo {a b : ℝ} : f.Measure (Ioo a b) = ofReal (leftLim f b - f a) := by @@ -604,6 +758,12 @@ theorem measure_Ioo {a b : ℝ} : f.Measure (Ioo a b) = ofReal (leftLim f b - f · simp only [f.mono.le_left_lim hab, sub_nonneg] #align stieltjes_function.measure_Ioo StieltjesFunction.measure_Ioo +/- warning: stieltjes_function.measure_Ico -> StieltjesFunction.measure_Ico is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Ico.{0} Real Real.preorder a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) b) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) a))) +but is expected to have type + forall (f : StieltjesFunction) (a : Real) (b : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Ico.{0} Real Real.instPreorderReal a b)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) b) (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) a))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Ico StieltjesFunction.measure_Icoₓ'. -/ @[simp] theorem measure_Ico (a b : ℝ) : f.Measure (Ico a b) = ofReal (leftLim f b - leftLim f a) := by @@ -616,6 +776,12 @@ theorem measure_Ico (a b : ℝ) : f.Measure (Ico a b) = ofReal (leftLim f b - le measure_union A measurableSet_Ioo, f.mono.le_left_lim hab, ← ENNReal.ofReal_add] #align stieltjes_function.measure_Ico StieltjesFunction.measure_Ico +/- warning: stieltjes_function.measure_Iic -> StieltjesFunction.measure_Iic is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {l : Real}, (Filter.Tendsto.{0, 0} Real Real (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) (Filter.atBot.{0} Real Real.preorder) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (forall (x : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Iic.{0} Real Real.preorder x)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f x) l))) +but is expected to have type + forall (f : StieltjesFunction) {l : Real}, (Filter.Tendsto.{0, 0} Real Real (StieltjesFunction.toFun f) (Filter.atBot.{0} Real Real.instPreorderReal) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (forall (x : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Iic.{0} Real Real.instPreorderReal x)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (StieltjesFunction.toFun f x) l))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Iic StieltjesFunction.measure_Iicₓ'. -/ theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) : f.Measure (Iic x) = ofReal (f x - l) := by @@ -624,6 +790,12 @@ theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) : exact ENNReal.tendsto_ofReal (tendsto.const_sub _ hf) #align stieltjes_function.measure_Iic StieltjesFunction.measure_Iic +/- warning: stieltjes_function.measure_Ici -> StieltjesFunction.measure_Ici is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {l : Real}, (Filter.Tendsto.{0, 0} Real Real (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) (Filter.atTop.{0} Real Real.preorder) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (forall (x : Real), Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.Ici.{0} Real Real.preorder x)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) l (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) x)))) +but is expected to have type + forall (f : StieltjesFunction) {l : Real}, (Filter.Tendsto.{0, 0} Real Real (StieltjesFunction.toFun f) (Filter.atTop.{0} Real Real.instPreorderReal) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (forall (x : Real), Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.Ici.{0} Real Real.instPreorderReal x)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) l (Function.leftLim.{0, 0} Real Real Real.linearOrder (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (StieltjesFunction.toFun f) x)))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_Ici StieltjesFunction.measure_Iciₓ'. -/ theorem measure_Ici {l : ℝ} (hf : Tendsto f atTop (𝓝 l)) (x : ℝ) : f.Measure (Ici x) = ofReal (l - leftLim f x) := by @@ -637,6 +809,12 @@ theorem measure_Ici {l : ℝ} (hf : Tendsto f atTop (𝓝 l)) (x : ℝ) : exact fun y => ⟨y + 1, fun z hyz => by rwa [le_sub_iff_add_le]⟩ #align stieltjes_function.measure_Ici StieltjesFunction.measure_Ici +/- warning: stieltjes_function.measure_univ -> StieltjesFunction.measure_univ is a dubious translation: +lean 3 declaration is + forall (f : StieltjesFunction) {l : Real} {u : Real}, (Filter.Tendsto.{0, 0} Real Real (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) (Filter.atBot.{0} Real Real.preorder) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (Filter.Tendsto.{0, 0} Real Real (coeFn.{1, 1} StieltjesFunction (fun (_x : StieltjesFunction) => Real -> Real) StieltjesFunction.instCoeFun f) (Filter.atTop.{0} Real Real.preorder) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) u)) -> (Eq.{1} ENNReal (coeFn.{1, 1} (MeasureTheory.Measure.{0} Real Real.measurableSpace) (fun (_x : MeasureTheory.Measure.{0} Real Real.measurableSpace) => (Set.{0} Real) -> ENNReal) (MeasureTheory.Measure.instCoeFun.{0} Real Real.measurableSpace) (StieltjesFunction.measure f) (Set.univ.{0} Real)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) u l))) +but is expected to have type + forall (f : StieltjesFunction) {l : Real} {u : Real}, (Filter.Tendsto.{0, 0} Real Real (StieltjesFunction.toFun f) (Filter.atBot.{0} Real Real.instPreorderReal) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) l)) -> (Filter.Tendsto.{0, 0} Real Real (StieltjesFunction.toFun f) (Filter.atTop.{0} Real Real.instPreorderReal) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) u)) -> (Eq.{1} ENNReal (MeasureTheory.OuterMeasure.measureOf.{0} Real (MeasureTheory.Measure.toOuterMeasure.{0} Real Real.measurableSpace (StieltjesFunction.measure f)) (Set.univ.{0} Real)) (ENNReal.ofReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) u l))) +Case conversion may be inaccurate. Consider using '#align stieltjes_function.measure_univ StieltjesFunction.measure_univₓ'. -/ theorem measure_univ {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) : f.Measure univ = ofReal (u - l) := by diff --git a/Mathbin/Probability/Integration.lean b/Mathbin/Probability/Integration.lean index 7e792d98ae..3fcd5ed96a 100644 --- a/Mathbin/Probability/Integration.lean +++ b/Mathbin/Probability/Integration.lean @@ -54,7 +54,7 @@ theorem lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : M revert f have h_mul_indicator : ∀ g, Measurable g → Measurable fun a => g a * T.indicator (fun x => c) a := fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T) - apply Measurable.eNNReal_induction + apply Measurable.ennreal_induction · intro c' s' h_meas_s' simp_rw [← inter_indicator_mul] rw [lintegral_indicator _ (MeasurableSet.inter (hMf _ h_meas_s') h_meas_T), @@ -92,7 +92,7 @@ theorem lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace by revert g have h_measM_f : Measurable f := h_meas_f.mono hMf le_rfl - apply Measurable.eNNReal_induction + apply Measurable.ennreal_induction · intro c s h_s apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f apply indep_sets_of_indep_sets_of_le_right h_ind diff --git a/Mathbin/RingTheory/Ideal/MinimalPrime.lean b/Mathbin/RingTheory/Ideal/MinimalPrime.lean index 5a078d944e..bb3c046130 100644 --- a/Mathbin/RingTheory/Ideal/MinimalPrime.lean +++ b/Mathbin/RingTheory/Ideal/MinimalPrime.lean @@ -37,19 +37,29 @@ section variable {R S : Type _} [CommRing R] [CommRing S] (I J : Ideal R) +#print Ideal.minimalPrimes /- /-- `I.minimal_primes` is the set of ideals that are minimal primes over `I`. -/ def Ideal.minimalPrimes : Set (Ideal R) := minimals (· ≤ ·) { p | p.IsPrime ∧ I ≤ p } #align ideal.minimal_primes Ideal.minimalPrimes +-/ +#print minimalPrimes /- /-- `minimal_primes R` is the set of minimal primes of `R`. This is defined as `ideal.minimal_primes ⊥`. -/ def minimalPrimes (R : Type _) [CommRing R] : Set (Ideal R) := Ideal.minimalPrimes ⊥ #align minimal_primes minimalPrimes +-/ variable {I J} +/- warning: ideal.exists_minimal_primes_le -> Ideal.exists_minimalPrimes_le is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [_inst_3 : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) J], (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) I J) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (p : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 I)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 I)) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p J))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} {J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} [_inst_3 : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) J], (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) I J) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (p : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 I)) (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) p J))) +Case conversion may be inaccurate. Consider using '#align ideal.exists_minimal_primes_le Ideal.exists_minimalPrimes_leₓ'. -/ theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by suffices @@ -75,6 +85,7 @@ theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.mi exact sInf_le hz #align ideal.exists_minimal_primes_le Ideal.exists_minimalPrimes_le +#print Ideal.radical_minimalPrimes /- @[simp] theorem Ideal.radical_minimalPrimes : I.radical.minimalPrimes = I.minimalPrimes := by @@ -83,7 +94,14 @@ theorem Ideal.radical_minimalPrimes : I.radical.minimalPrimes = I.minimalPrimes ext p exact ⟨fun ⟨a, b⟩ => ⟨a, ideal.le_radical.trans b⟩, fun ⟨a, b⟩ => ⟨a, a.radical_le_iff.mpr b⟩⟩ #align ideal.radical_minimal_primes Ideal.radical_minimalPrimes +-/ +/- warning: ideal.Inf_minimal_primes -> Ideal.sInf_minimalPrimes is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (InfSet.sInf.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasInf.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.minimalPrimes.{u1} R _inst_1 I)) (Ideal.radical.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) I) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (InfSet.sInf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.minimalPrimes.{u1} R _inst_1 I)) (Ideal.radical.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) I) +Case conversion may be inaccurate. Consider using '#align ideal.Inf_minimal_primes Ideal.sInf_minimalPrimesₓ'. -/ @[simp] theorem Ideal.sInf_minimalPrimes : sInf I.minimalPrimes = I.radical := by @@ -100,6 +118,12 @@ theorem Ideal.sInf_minimalPrimes : sInf I.minimalPrimes = I.radical := exact hI.1.symm #align ideal.Inf_minimal_primes Ideal.sInf_minimalPrimes +/- warning: ideal.exists_comap_eq_of_mem_minimal_primes_of_injective -> Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))}, (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f)) -> (forall (p : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p (minimalPrimes.{u1} R _inst_1)) -> (Exists.{succ u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (fun (p' : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) => And (Ideal.IsPrime.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) p') (Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f p') p)))) +but is expected to have type + forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} S] {f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))}, (Function.Injective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) f)) -> (forall (p : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), (Membership.mem.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Set.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Set.instMembershipSet.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) p (minimalPrimes.{u2} R _inst_1)) -> (Exists.{succ u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (fun (p' : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) => And (Ideal.IsPrime.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) p') (Eq.{succ u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) f p') p)))) +Case conversion may be inaccurate. Consider using '#align ideal.exists_comap_eq_of_mem_minimal_primes_of_injective Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injectiveₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (p «expr ∈ » minimal_primes[minimal_primes] R) -/ theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S} (hf : Function.Injective f) (p) (_ : p ∈ minimalPrimes R) : @@ -131,6 +155,12 @@ theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S} infer_instance #align ideal.exists_comap_eq_of_mem_minimal_primes_of_injective Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective +/- warning: ideal.exists_comap_eq_of_mem_minimal_primes -> Ideal.exists_comap_eq_of_mem_minimalPrimes is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {I : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))} (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (p : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f I))) -> (Exists.{succ u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (fun (p' : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) => And (Ideal.IsPrime.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) p') (And (LE.le.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Preorder.toHasLe.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (CompleteSemilatticeInf.toPartialOrder.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Submodule.completeLattice.{u2, u2} S S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) I p') (Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f p') p)))) +but is expected to have type + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {I : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))} (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (p : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) f I))) -> (Exists.{succ u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (fun (p' : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) => And (Ideal.IsPrime.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) p') (And (LE.le.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Preorder.toLE.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Submodule.completeLattice.{u2, u2} S S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (Semiring.toModule.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))))))) I p') (Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) f p') p)))) +Case conversion may be inaccurate. Consider using '#align ideal.exists_comap_eq_of_mem_minimal_primes Ideal.exists_comap_eq_of_mem_minimalPrimesₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (p «expr ∈ » (I.comap f).minimal_primes) -/ theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p) (_ : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p := @@ -171,6 +201,12 @@ theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) exacts[sup_le rfl.le this, Ideal.Quotient.mk_surjective] #align ideal.exists_comap_eq_of_mem_minimal_primes Ideal.exists_comap_eq_of_mem_minimalPrimes +/- warning: ideal.exists_minimal_primes_comap_eq -> Ideal.exists_minimalPrimes_comap_eq is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {I : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))} (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (p : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f I))) -> (Exists.{succ u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (fun (p' : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) => Exists.{0} (Membership.Mem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Set.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Set.hasMem.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) p' (Ideal.minimalPrimes.{u2} S _inst_2 I)) (fun (H : Membership.Mem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Set.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Set.hasMem.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) p' (Ideal.minimalPrimes.{u2} S _inst_2 I)) => Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f p') p))) +but is expected to have type + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {I : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))} (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (p : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) f I))) -> (Exists.{succ u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (fun (p' : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) => And (Membership.mem.{u2, u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Set.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (Set.instMembershipSet.{u2} (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) p' (Ideal.minimalPrimes.{u2} S _inst_2 I)) (Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) f p') p))) +Case conversion may be inaccurate. Consider using '#align ideal.exists_minimal_primes_comap_eq Ideal.exists_minimalPrimes_comap_eqₓ'. -/ /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (p «expr ∈ » (I.comap f).minimal_primes) -/ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p) (_ : p ∈ (I.comap f).minimalPrimes) : ∃ p' ∈ I.minimalPrimes, Ideal.comap f p' = p := @@ -184,6 +220,12 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p) exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.1.2⟩ this).antisymm this #align ideal.exists_minimal_primes_comap_eq Ideal.exists_minimalPrimes_comap_eq +/- warning: ideal.mimimal_primes_comap_of_surjective -> Ideal.mimimal_primes_comap_of_surjective is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))}, (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f)) -> (forall {I : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))} {J : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))}, (Membership.Mem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Set.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Set.hasMem.{u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) J (Ideal.minimalPrimes.{u2} S _inst_2 I)) -> (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f J) (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f I)))) +but is expected to have type + forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} S] {f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))}, (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) f)) -> (forall {I : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))} {J : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))}, (Membership.mem.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (Set.{u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (Set.instMembershipSet.{u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) J (Ideal.minimalPrimes.{u1} S _inst_2 I)) -> (Membership.mem.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Set.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Set.instMembershipSet.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) f J) (Ideal.minimalPrimes.{u2} R _inst_1 (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) f I)))) +Case conversion may be inaccurate. Consider using '#align ideal.mimimal_primes_comap_of_surjective Ideal.mimimal_primes_comap_of_surjectiveₓ'. -/ theorem Ideal.mimimal_primes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by @@ -198,6 +240,12 @@ theorem Ideal.mimimal_primes_comap_of_surjective {f : R →+* S} (hf : Function. · exact Ideal.map_le_of_le_comap e₂ #align ideal.mimimal_primes_comap_of_surjective Ideal.mimimal_primes_comap_of_surjective +/- warning: ideal.comap_minimal_primes_eq_of_surjective -> Ideal.comap_minimalPrimes_eq_of_surjective is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))}, (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f)) -> (forall (I : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))), Eq.{succ u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.minimalPrimes.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f I)) (Set.image.{u2, u1} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f) (Ideal.minimalPrimes.{u2} S _inst_2 I))) +but is expected to have type + forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} S] {f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))}, (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) f)) -> (forall (I : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))), Eq.{succ u2} (Set.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Ideal.minimalPrimes.{u2} R _inst_1 (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) f I)) (Set.image.{u1, u2} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) f) (Ideal.minimalPrimes.{u1} S _inst_2 I))) +Case conversion may be inaccurate. Consider using '#align ideal.comap_minimal_primes_eq_of_surjective Ideal.comap_minimalPrimes_eq_of_surjectiveₓ'. -/ theorem Ideal.comap_minimalPrimes_eq_of_surjective {f : R →+* S} (hf : Function.Surjective f) (I : Ideal S) : (I.comap f).minimalPrimes = Ideal.comap f '' I.minimalPrimes := by @@ -210,12 +258,19 @@ theorem Ideal.comap_minimalPrimes_eq_of_surjective {f : R →+* S} (hf : Functio exact Ideal.mimimal_primes_comap_of_surjective hf hJ #align ideal.comap_minimal_primes_eq_of_surjective Ideal.comap_minimalPrimes_eq_of_surjective +/- warning: ideal.minimal_primes_eq_comap -> Ideal.minimalPrimes_eq_comap is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Eq.{succ u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.minimalPrimes.{u1} R _inst_1 I) (Set.image.{u1, u1} (Ideal.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.comap.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (RingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (RingHom.ringHomClass.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (minimalPrimes.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, Eq.{succ u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.minimalPrimes.{u1} R _inst_1 I) (Set.image.{u1, u1} (Ideal.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R 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(Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (RingHom.instRingHomClassRingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (minimalPrimes.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) +Case conversion may be inaccurate. Consider using '#align ideal.minimal_primes_eq_comap Ideal.minimalPrimes_eq_comapₓ'. -/ theorem Ideal.minimalPrimes_eq_comap : I.minimalPrimes = Ideal.comap I.Quotient.mk '' minimalPrimes (R ⧸ I) := by rw [minimalPrimes, ← Ideal.comap_minimalPrimes_eq_of_surjective Ideal.Quotient.mk_surjective, ← RingHom.ker_eq_comap_bot, Ideal.mk_ker] #align ideal.minimal_primes_eq_comap Ideal.minimalPrimes_eq_comap +#print Ideal.minimalPrimes_eq_subsingleton /- theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes = {I.radical} := by ext J @@ -227,7 +282,9 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes · rintro (rfl : J = I.radical) exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩ #align ideal.minimal_primes_eq_subsingleton Ideal.minimalPrimes_eq_subsingleton +-/ +#print Ideal.minimalPrimes_eq_subsingleton_self /- theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = {I} := by ext J @@ -236,6 +293,7 @@ theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = · rintro (rfl : J = I) refine' ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩ #align ideal.minimal_primes_eq_subsingleton_self Ideal.minimalPrimes_eq_subsingleton_self +-/ end diff --git a/Mathbin/RingTheory/Localization/AtPrime.lean b/Mathbin/RingTheory/Localization/AtPrime.lean index 5609ab7c8a..b1882efb57 100644 --- a/Mathbin/RingTheory/Localization/AtPrime.lean +++ b/Mathbin/RingTheory/Localization/AtPrime.lean @@ -340,7 +340,7 @@ theorem localRingHom_to_map (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P) (hIJ lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {P : Type.{u2}} [_inst_4 : CommSemiring.{u2} P] (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [hI : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) I] (J : Ideal.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4)) [hJ : Ideal.IsPrime.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4) J] (f : RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (hIJ : Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) I (Ideal.comap.{u1, u2, max u1 u2} R P (RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} P _inst_4) (RingHom.ringHomClass.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) f J)) (x : R) (y : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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(Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (hIJ : Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) I (Ideal.comap.{u1, u2, max u1 u2} R P (RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} P _inst_4) (RingHom.instRingHomClassRingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) f J)) (x : R) (y : Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x (Ideal.primeCompl.{u1} R _inst_1 I hI))), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Localization.AtPrime.{u1} R _inst_1 I hI) => Localization.AtPrime.{u2} P _inst_4 J hJ) (IsLocalization.mk'.{u1, u1} R _inst_1 (Ideal.primeCompl.{u1} R _inst_1 I hI) (Localization.AtPrime.{u1} R _inst_1 I hI) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 (Ideal.primeCompl.{u1} R _inst_1 I hI)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 (Ideal.primeCompl.{u1} R _inst_1 I hI) R _inst_1 (Algebra.id.{u1} R _inst_1)) 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(Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))) (Ideal.comap.{u1, u2, max u1 u2} R P (RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} P _inst_4) (RingHom.instRingHomClassRingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) f J) I) (Localization.le_comap_primeCompl_iff.{u1, u2} R _inst_1 P _inst_4 I hI J hJ f) (ge_of_eq.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) I (Ideal.comap.{u1, u2, max u1 u2} R P (RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} P _inst_4) (RingHom.instRingHomClassRingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) f J) hIJ) (Subtype.val.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x (SetLike.coe.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Ideal.primeCompl.{u1} R _inst_1 I hI))) y) (Subtype.property.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x (Ideal.primeCompl.{u1} R _inst_1 I hI)) y)))) Case conversion may be inaccurate. Consider using '#align localization.local_ring_hom_mk' Localization.localRingHom_mk'ₓ'. -/ theorem localRingHom_mk' (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R) (y : I.primeCompl) : diff --git a/Mathbin/RingTheory/Localization/Basic.lean b/Mathbin/RingTheory/Localization/Basic.lean index 494937842f..448151390b 100644 --- a/Mathbin/RingTheory/Localization/Basic.lean +++ b/Mathbin/RingTheory/Localization/Basic.lean @@ -1593,7 +1593,7 @@ end lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (Submonoid.LocalizationMap.{u1, u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (CommSemiring.toCommMonoid.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M))) (IsLocalization.toLocalizationMap.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M)) (Localization.monoidOf.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) but is expected to have type - forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (Submonoid.LocalizationMap.{u1, u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (CommSemiring.toCommMonoid.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M))) (IsLocalization.toLocalizationMap.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.instIsLocalizationLocalizationToCommMonoidInstCommSemiringLocalizationToCommMonoidInstAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoidId.{u1} R _inst_1 M)) (Localization.monoidOf.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) + forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (Submonoid.LocalizationMap.{u1, u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (CommSemiring.toCommMonoid.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M))) (IsLocalization.toLocalizationMap.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M)) (Localization.monoidOf.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) Case conversion may be inaccurate. Consider using '#align localization.to_localization_map_eq_monoid_of Localization.toLocalizationMap_eq_monoidOfₓ'. -/ @[simp] theorem toLocalizationMap_eq_monoidOf : toLocalizationMap M (Localization M) = monoidOf M := @@ -1624,7 +1624,7 @@ theorem mk_one_eq_algebraMap (x) : mk x 1 = algebraMap R (Localization M) x := lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))} (x : R) (y : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) M), Eq.{succ u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M x y) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M) x y) but is expected to have type - forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))} (x : R) (y : Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) x M)), Eq.{succ u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M x y) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.instIsLocalizationLocalizationToCommMonoidInstCommSemiringLocalizationToCommMonoidInstAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoidId.{u1} R _inst_1 M) x y) + forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))} (x : R) (y : Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) x M)), Eq.{succ u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M x y) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M) x y) Case conversion may be inaccurate. Consider using '#align localization.mk_eq_mk'_apply Localization.mk_eq_mk'_applyₓ'. -/ theorem mk_eq_mk'_apply (x y) : mk x y = IsLocalization.mk' (Localization M) x y := by rw [mk_eq_monoid_of_mk'_apply, mk', to_localization_map_eq_monoid_of] @@ -1634,7 +1634,7 @@ theorem mk_eq_mk'_apply (x y) : mk x y = IsLocalization.mk' (Localization M) x y lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (R -> (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) M) -> (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M)) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M)) but is expected to have type - forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (R -> (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) x M)) -> (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M)) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.instIsLocalizationLocalizationToCommMonoidInstCommSemiringLocalizationToCommMonoidInstAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoidId.{u1} R _inst_1 M)) + forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))}, Eq.{succ u1} (R -> (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)))) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1))))) x M)) -> (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M)) (Localization.mk.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M)) Case conversion may be inaccurate. Consider using '#align localization.mk_eq_mk' Localization.mk_eq_mk'ₓ'. -/ @[simp] theorem mk_eq_mk' : (mk : R → M → Localization M) = IsLocalization.mk' (Localization M) := @@ -1695,7 +1695,7 @@ end lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))} {S : Type.{u2}} [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : IsLocalization.{u1, u2} R _inst_1 M S _inst_2 _inst_3] (x : R) (y : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M), Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (AlgEquiv.{u1, u1, u2} R (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) (CommSemiring.toSemiring.{u2} S _inst_2) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) _inst_3) (fun (_x : AlgEquiv.{u1, u1, u2} R (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) (CommSemiring.toSemiring.{u2} S _inst_2) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) _inst_3) => (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) -> S) (AlgEquiv.hasCoeToFun.{u1, u1, u2} R (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) (CommSemiring.toSemiring.{u2} S _inst_2) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) _inst_3) (Localization.algEquiv.{u1, u2} R _inst_1 M S _inst_2 _inst_3 _inst_5) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M) x y)) (IsLocalization.mk'.{u1, u2} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y) but is expected to have type - forall {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))} (S : Type.{u1}) [_inst_2 : CommSemiring.{u1} S] [_inst_3 : Algebra.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2)] [_inst_5 : IsLocalization.{u2, u1} R _inst_1 M S _inst_2 _inst_3] (x : R) (y : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))))) x M)), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) => S) (IsLocalization.mk'.{u2, u2} R _inst_1 M (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (Localization.instIsLocalizationLocalizationToCommMonoidInstCommSemiringLocalizationToCommMonoidInstAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoidId.{u2} R _inst_1 M) x y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (AlgEquiv.{u2, u2, u1} R 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u1, u2, u2, u1} (AlgEquiv.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (CommSemiring.toSemiring.{u1} S _inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3) R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3)) (SemilinearMapClass.distribMulActionHomClass.{u2, u2, u1, max u2 u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S (AlgEquiv.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (CommSemiring.toSemiring.{u1} S _inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3) (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) 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(Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3 (AlgEquiv.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (CommSemiring.toSemiring.{u1} S _inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3) (AlgEquivClass.toAlgHomClass.{max u2 u1, u2, u2, u1} (AlgEquiv.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) 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_inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3)))))) (Localization.algEquiv.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5) (IsLocalization.mk'.{u2, u2} R _inst_1 M (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (Localization.isLocalization.{u2} R _inst_1 M) x y)) (IsLocalization.mk'.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y) Case conversion may be inaccurate. Consider using '#align localization.alg_equiv_mk' Localization.algEquiv_mk'ₓ'. -/ @[simp] theorem algEquiv_mk' (x : R) (y : M) : algEquiv M S (mk' (Localization M) x y) = mk' S x y := @@ -1706,7 +1706,7 @@ theorem algEquiv_mk' (x : R) (y : M) : algEquiv M S (mk' (Localization M) x y) = lean 3 declaration is forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))} {S : Type.{u2}} [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : IsLocalization.{u1, u2} R _inst_1 M S _inst_2 _inst_3] (x : R) (y : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M), Eq.{succ u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (AlgEquiv.{u1, u2, u1} R S (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) _inst_3 (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1))) (fun (_x : AlgEquiv.{u1, u2, u1} R S (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) _inst_3 (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1))) => S -> (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M)) (AlgEquiv.hasCoeToFun.{u1, u2, u1} R S (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) _inst_3 (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1))) (AlgEquiv.symm.{u1, u1, u2} R (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M)) (CommSemiring.toSemiring.{u2} S _inst_2) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) _inst_3 (Localization.algEquiv.{u1, u2} R _inst_1 M S _inst_2 _inst_3 _inst_5)) (IsLocalization.mk'.{u1, u2} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y)) (IsLocalization.mk'.{u1, u1} R _inst_1 M (Localization.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) M) (Localization.commSemiring.{u1} R _inst_1 M) (Localization.algebra.{u1, u1} R _inst_1 M R _inst_1 (Algebra.id.{u1} R _inst_1)) (Localization.isLocalization.{u1} R _inst_1 M) x y) but is expected to have type - forall {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))} (S : Type.{u1}) [_inst_2 : CommSemiring.{u1} S] [_inst_3 : Algebra.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2)] [_inst_5 : IsLocalization.{u2, u1} R _inst_1 M S _inst_2 _inst_3] (x : R) (y : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))))) x M)), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : S) => Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (IsLocalization.mk'.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y)) (FunLike.coe.{max (succ u2) 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2))))) (AddCommMonoid.toAddMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3)) (Module.toDistribMulAction.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) (AlgHomClass.linearMapClass.{u2, u1, u2, max u2 u1} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) (AlgEquivClass.toAlgHomClass.{max u2 u1, u2, u1, u2} (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (AlgEquiv.instAlgEquivClassAlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)))))))) (AlgEquiv.symm.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (CommSemiring.toSemiring.{u1} S _inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3 (Localization.algEquiv.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5)) (IsLocalization.mk'.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y)) (IsLocalization.mk'.{u2, u2} R _inst_1 M (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (Localization.instIsLocalizationLocalizationToCommMonoidInstCommSemiringLocalizationToCommMonoidInstAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoidId.{u2} R _inst_1 M) x y) + forall {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))} (S : Type.{u1}) [_inst_2 : CommSemiring.{u1} S] [_inst_3 : Algebra.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2)] [_inst_5 : IsLocalization.{u2, u1} R _inst_1 M S _inst_2 _inst_3] (x : R) (y : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))))) x M)), 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(CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u1, u2, u1, u2} (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2))))) (AddCommMonoid.toAddMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3)) (Module.toDistribMulAction.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)))) (SemilinearMapClass.distribMulActionHomClass.{u2, u1, u2, max u2 u1} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S _inst_2)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Semiring.toNonAssocSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M))))) (Algebra.toModule.{u2, u1} R S _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) _inst_3) (Algebra.toModule.{u2, u2} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) (AlgHomClass.linearMapClass.{u2, u1, u2, max u2 u1} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) (AlgEquivClass.toAlgHomClass.{max u2 u1, u2, u1, u2} (AlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1))) R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (AlgEquiv.instAlgEquivClassAlgEquiv.{u2, u1, u2} R S (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) _inst_1 (CommSemiring.toSemiring.{u1} S _inst_2) (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) _inst_3 (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)))))))) (AlgEquiv.symm.{u2, u2, u1} R (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) S _inst_1 (CommSemiring.toSemiring.{u2} (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M)) (CommSemiring.toSemiring.{u1} S _inst_2) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) _inst_3 (Localization.algEquiv.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5)) (IsLocalization.mk'.{u2, u1} R _inst_1 M S _inst_2 _inst_3 _inst_5 x y)) (IsLocalization.mk'.{u2, u2} R _inst_1 M (Localization.{u2} R (CommSemiring.toCommMonoid.{u2} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u2} R _inst_1 M) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u2, u2} R _inst_1 M R _inst_1 (Algebra.id.{u2} R _inst_1)) (Localization.isLocalization.{u2} R _inst_1 M) x y) Case conversion may be inaccurate. Consider using '#align localization.alg_equiv_symm_mk' Localization.algEquiv_symm_mk'ₓ'. -/ @[simp] theorem algEquiv_symm_mk' (x : R) (y : M) : diff --git a/Mathbin/Topology/Algebra/Module/LocallyConvex.lean b/Mathbin/Topology/Algebra/Module/LocallyConvex.lean index 87a254f6de..2385f79a59 100644 --- a/Mathbin/Topology/Algebra/Module/LocallyConvex.lean +++ b/Mathbin/Topology/Algebra/Module/LocallyConvex.lean @@ -40,20 +40,29 @@ open Topology Pointwise section Semimodule +#print LocallyConvexSpace /- /-- A `locally_convex_space` is a topological semimodule over an ordered semiring in which convex neighborhoods of a point form a neighborhood basis at that point. -/ class LocallyConvexSpace (𝕜 E : Type _) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Prop where convex_basis : ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id #align locally_convex_space LocallyConvexSpace +-/ variable (𝕜 E : Type _) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] +/- warning: locally_convex_space_iff -> locallyConvexSpace_iff is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u2} E], Iff (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4) (forall (x : E), Filter.HasBasis.{u2, succ u2} E (Set.{u2} E) (nhds.{u2} E _inst_4 x) (fun (s : Set.{u2} E) => And (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) s (nhds.{u2} E _inst_4 x)) (Convex.{u1, u2} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2 _inst_3)))) s)) (id.{succ u2} (Set.{u2} E))) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u1} E], Iff (LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4) (forall (x : E), Filter.HasBasis.{u1, succ u1} E (Set.{u1} E) (nhds.{u1} E _inst_4 x) (fun (s : Set.{u1} E) => And (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) s (nhds.{u1} E _inst_4 x)) (Convex.{u2, u1} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (MonoidWithZero.toZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2 _inst_3)))) s)) (id.{succ u1} (Set.{u1} E))) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_iff locallyConvexSpace_iffₓ'. -/ theorem locallyConvexSpace_iff : LocallyConvexSpace 𝕜 E ↔ ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id := ⟨@LocallyConvexSpace.convex_basis _ _ _ _ _ _, LocallyConvexSpace.mk⟩ #align locally_convex_space_iff locallyConvexSpace_iff +#print LocallyConvexSpace.ofBases /- theorem LocallyConvexSpace.ofBases {ι : Type _} (b : E → ι → Set E) (p : E → ι → Prop) (hbasis : ∀ x : E, (𝓝 x).HasBasis (p x) (b x)) (hconvex : ∀ x i, p x i → Convex 𝕜 (b x i)) : LocallyConvexSpace 𝕜 E := @@ -63,12 +72,25 @@ theorem LocallyConvexSpace.ofBases {ι : Type _} (b : E → ι → Set E) (p : E fun s hs => ⟨(hbasis x).index s hs.1, ⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩⟩ #align locally_convex_space.of_bases LocallyConvexSpace.ofBases +-/ +/- warning: locally_convex_space.convex_basis_zero -> LocallyConvexSpace.convex_basis_zero is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4], Filter.HasBasis.{u2, succ u2} E (Set.{u2} E) (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))))))) (fun (s : Set.{u2} E) => And (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) s (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2)))))))) (Convex.{u1, u2} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2 _inst_3)))) s)) (id.{succ u2} (Set.{u2} E)) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4], Filter.HasBasis.{u1, succ u1} E (Set.{u1} E) (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2))))) (fun (s : Set.{u1} E) => And (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) s (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)))))) (Convex.{u2, u1} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (MonoidWithZero.toZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2 _inst_3)))) s)) (id.{succ u1} (Set.{u1} E)) +Case conversion may be inaccurate. Consider using '#align locally_convex_space.convex_basis_zero LocallyConvexSpace.convex_basis_zeroₓ'. -/ theorem LocallyConvexSpace.convex_basis_zero [LocallyConvexSpace 𝕜 E] : (𝓝 0 : Filter E).HasBasis (fun s => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id := LocallyConvexSpace.convex_basis 0 #align locally_convex_space.convex_basis_zero LocallyConvexSpace.convex_basis_zero +/- warning: locally_convex_space_iff_exists_convex_subset -> locallyConvexSpace_iff_exists_convex_subset is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u2} E], Iff (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4) (forall (x : E) (U : Set.{u2} E), (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) U (nhds.{u2} E _inst_4 x)) -> (Exists.{succ u2} (Set.{u2} E) (fun (S : Set.{u2} E) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) S (nhds.{u2} E _inst_4 x)) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) S (nhds.{u2} E _inst_4 x)) => And (Convex.{u1, u2} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E _inst_2))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2 _inst_3)))) S) (HasSubset.Subset.{u2} (Set.{u2} E) (Set.hasSubset.{u2} E) S U))))) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] [_inst_4 : TopologicalSpace.{u1} E], Iff (LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 _inst_2 _inst_3 _inst_4) (forall (x : E) (U : Set.{u1} E), (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) U (nhds.{u1} E _inst_4 x)) -> (Exists.{succ u1} (Set.{u1} E) (fun (S : Set.{u1} E) => And (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) S (nhds.{u1} E _inst_4 x)) (And (Convex.{u2, u1} 𝕜 E _inst_1 _inst_2 (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (MonoidWithZero.toZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)) (AddMonoid.toZero.{u1} E (AddCommMonoid.toAddMonoid.{u1} E _inst_2)) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2 _inst_3)))) S) (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) S U))))) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_iff_exists_convex_subset locallyConvexSpace_iff_exists_convex_subsetₓ'. -/ theorem locallyConvexSpace_iff_exists_convex_subset : LocallyConvexSpace 𝕜 E ↔ ∀ x : E, ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U := (locallyConvexSpace_iff 𝕜 E).trans (forall_congr' fun x => hasBasis_self) @@ -81,6 +103,12 @@ section Module variable (𝕜 E : Type _) [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] +/- warning: locally_convex_space.of_basis_zero -> LocallyConvexSpace.ofBasisZero is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)] {ι : Type.{u3}} (b : ι -> (Set.{u2} E)) (p : ι -> Prop), (Filter.HasBasis.{u2, succ u3} E ι (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2))))))))) p b) -> (forall (i : ι), (p i) -> (Convex.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (b i))) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4) +but is expected to have type + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)] {ι : Type.{u3}} (b : ι -> (Set.{u2} E)) (p : ι -> Prop), (Filter.HasBasis.{u2, succ u3} E ι (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (Zero.toOfNat0.{u2} E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2)))))))) p b) -> (forall (i : ι), (p i) -> (Convex.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (MonoidWithZero.toZero.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_2))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) (b i))) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4) +Case conversion may be inaccurate. Consider using '#align locally_convex_space.of_basis_zero LocallyConvexSpace.ofBasisZeroₓ'. -/ theorem LocallyConvexSpace.ofBasisZero {ι : Type _} (b : ι → Set E) (p : ι → Prop) (hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ i, p i → Convex 𝕜 (b i)) : LocallyConvexSpace 𝕜 E := by @@ -91,6 +119,12 @@ theorem LocallyConvexSpace.ofBasisZero {ι : Type _} (b : ι → Set E) (p : ι exact hbasis.map _ #align locally_convex_space.of_basis_zero LocallyConvexSpace.ofBasisZero +/- warning: locally_convex_space_iff_zero -> locallyConvexSpace_iff_zero is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)], Iff (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4) (Filter.HasBasis.{u2, succ u2} E (Set.{u2} E) (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2))))))))) (fun (s : Set.{u2} E) => And (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) s (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))))))) (Convex.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) s)) (id.{succ u2} (Set.{u2} E))) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : TopologicalAddGroup.{u1} E _inst_4 (AddCommGroup.toAddGroup.{u1} E _inst_2)], Iff (LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3 _inst_4) (Filter.HasBasis.{u1, succ u1} E (Set.{u1} E) (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2)))))))) (fun (s : Set.{u1} E) => And (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) s (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))))))) (Convex.{u2, u1} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (MonoidWithZero.toZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) s)) (id.{succ u1} (Set.{u1} E))) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_iff_zero locallyConvexSpace_iff_zeroₓ'. -/ theorem locallyConvexSpace_iff_zero : LocallyConvexSpace 𝕜 E ↔ (𝓝 0 : Filter E).HasBasis (fun s : Set E => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id := @@ -98,17 +132,25 @@ theorem locallyConvexSpace_iff_zero : LocallyConvexSpace.ofBasisZero 𝕜 E _ _ h fun s => And.right⟩ #align locally_convex_space_iff_zero locallyConvexSpace_iff_zero +/- warning: locally_convex_space_iff_exists_convex_subset_zero -> locallyConvexSpace_iff_exists_convex_subset_zero is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)], Iff (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4) (forall (U : Set.{u2} E), (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) U (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))))))) -> (Exists.{succ u2} (Set.{u2} E) (fun (S : Set.{u2} E) => Exists.{0} (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) S (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))))))) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) S (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))))))) => And (Convex.{u1, u2} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) S) (HasSubset.Subset.{u2} (Set.{u2} E) (Set.hasSubset.{u2} E) S U))))) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : TopologicalAddGroup.{u1} E _inst_4 (AddCommGroup.toAddGroup.{u1} E _inst_2)], Iff (LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3 _inst_4) (forall (U : Set.{u1} E), (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) U (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))))))) -> (Exists.{succ u1} (Set.{u1} E) (fun (S : Set.{u1} E) => And (Membership.mem.{u1, u1} (Set.{u1} E) (Filter.{u1} E) (instMembershipSetFilter.{u1} E) S (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))))))) (And (Convex.{u2, u1} 𝕜 E _inst_1 (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (MonoidWithZero.toZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) S) (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) S U))))) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_iff_exists_convex_subset_zero locallyConvexSpace_iff_exists_convex_subset_zeroₓ'. -/ theorem locallyConvexSpace_iff_exists_convex_subset_zero : LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ (𝓝 0 : Filter E), ∃ S ∈ (𝓝 0 : Filter E), Convex 𝕜 S ∧ S ⊆ U := (locallyConvexSpace_iff_zero 𝕜 E).trans hasBasis_self #align locally_convex_space_iff_exists_convex_subset_zero locallyConvexSpace_iff_exists_convex_subset_zero +#print LocallyConvexSpace.toLocallyConnectedSpace /- -- see Note [lower instance priority] -instance (priority := 100) LocallyConvexSpace.to_locallyConnectedSpace [Module ℝ E] +instance (priority := 100) LocallyConvexSpace.toLocallyConnectedSpace [Module ℝ E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] : LocallyConnectedSpace E := locallyConnectedSpace_of_connected_bases _ _ (fun x => @LocallyConvexSpace.convex_basis ℝ _ _ _ _ _ _ x) fun x s hs => hs.2.IsPreconnected -#align locally_convex_space.to_locally_connected_space LocallyConvexSpace.to_locallyConnectedSpace +#align locally_convex_space.to_locally_connected_space LocallyConvexSpace.toLocallyConnectedSpace +-/ end Module @@ -117,6 +159,12 @@ section LinearOrderedField variable (𝕜 E : Type _) [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] +/- warning: locally_convex_space.convex_open_basis_zero -> LocallyConvexSpace.convex_open_basis_zero is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) (E : Type.{u2}) [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)] [_inst_6 : ContinuousConstSMul.{u1, u2} 𝕜 E _inst_4 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3))))] [_inst_7 : LocallyConvexSpace.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4], Filter.HasBasis.{u2, succ u2} E (Set.{u2} E) (nhds.{u2} E _inst_4 (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2))))))))) (fun (s : Set.{u2} E) => And (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_2)))))))) s) (And (IsOpen.{u2} E _inst_4 s) (Convex.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) s))) (id.{succ u2} (Set.{u2} E)) +but is expected to have type + forall (𝕜 : Type.{u2}) (E : Type.{u1}) [_inst_1 : LinearOrderedField.{u2} 𝕜] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : TopologicalAddGroup.{u1} E _inst_4 (AddCommGroup.toAddGroup.{u1} E _inst_2)] [_inst_6 : ContinuousConstSMul.{u2, u1} 𝕜 E _inst_4 (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3))))] [_inst_7 : LocallyConvexSpace.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3 _inst_4], Filter.HasBasis.{u1, succ u1} E (Set.{u1} E) (nhds.{u1} E _inst_4 (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2)))))))) (fun (s : Set.{u1} E) => And (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))))) s) (And (IsOpen.{u1} E _inst_4 s) (Convex.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) s))) (id.{succ u1} (Set.{u1} E)) +Case conversion may be inaccurate. Consider using '#align locally_convex_space.convex_open_basis_zero LocallyConvexSpace.convex_open_basis_zeroₓ'. -/ theorem LocallyConvexSpace.convex_open_basis_zero [LocallyConvexSpace 𝕜 E] : (𝓝 0 : Filter E).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id := (LocallyConvexSpace.convex_basis_zero 𝕜 E).to_hasBasis @@ -128,6 +176,12 @@ theorem LocallyConvexSpace.convex_open_basis_zero [LocallyConvexSpace 𝕜 E] : variable {𝕜 E} +/- warning: disjoint.exists_open_convexes -> Disjoint.exists_open_convexes is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : AddCommGroup.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)] [_inst_4 : TopologicalSpace.{u2} E] [_inst_5 : TopologicalAddGroup.{u2} E _inst_4 (AddCommGroup.toAddGroup.{u2} E _inst_2)] [_inst_6 : ContinuousConstSMul.{u1, u2} 𝕜 E _inst_4 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3))))] [_inst_7 : LocallyConvexSpace.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3 _inst_4] {s : Set.{u2} E} {t : Set.{u2} E}, (Disjoint.{u2} (Set.{u2} E) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} E) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} E) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} E) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} E) (Set.completeBooleanAlgebra.{u2} E)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} E) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} E) (Set.booleanAlgebra.{u2} E))) s t) -> (Convex.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) s) -> (IsCompact.{u2} E _inst_4 s) -> (Convex.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) t) -> (IsClosed.{u2} E _inst_4 t) -> (Exists.{succ u2} (Set.{u2} E) (fun (u : Set.{u2} E) => Exists.{succ u2} (Set.{u2} E) (fun (v : Set.{u2} E) => And (IsOpen.{u2} E _inst_4 u) (And (IsOpen.{u2} E _inst_4 v) (And (Convex.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) u) (And (Convex.{u1, u2} 𝕜 E (StrictOrderedSemiring.toOrderedSemiring.{u1} 𝕜 (StrictOrderedRing.toStrictOrderedSemiring.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_2)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_2) _inst_3)))) v) (And (HasSubset.Subset.{u2} (Set.{u2} E) (Set.hasSubset.{u2} E) s u) (And (HasSubset.Subset.{u2} (Set.{u2} E) (Set.hasSubset.{u2} E) t v) (Disjoint.{u2} (Set.{u2} E) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} E) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} E) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} E) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} E) (Set.completeBooleanAlgebra.{u2} E)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} E) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} E) (Set.booleanAlgebra.{u2} E))) u v))))))))) +but is expected to have type + forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_1 : LinearOrderedField.{u2} 𝕜] [_inst_2 : AddCommGroup.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2)] [_inst_4 : TopologicalSpace.{u1} E] [_inst_5 : TopologicalAddGroup.{u1} E _inst_4 (AddCommGroup.toAddGroup.{u1} E _inst_2)] [_inst_6 : ContinuousConstSMul.{u2, u1} 𝕜 E _inst_4 (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3))))] [_inst_7 : LocallyConvexSpace.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3 _inst_4] {s : Set.{u1} E} {t : Set.{u1} E}, (Disjoint.{u1} (Set.{u1} E) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} E) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} E) (Preorder.toLE.{u1} (Set.{u1} E) (PartialOrder.toPreorder.{u1} (Set.{u1} E) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} E) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))) s t) -> (Convex.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) s) -> (IsCompact.{u1} E _inst_4 s) -> (Convex.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) t) -> (IsClosed.{u1} E _inst_4 t) -> (Exists.{succ u1} (Set.{u1} E) (fun (u : Set.{u1} E) => Exists.{succ u1} (Set.{u1} E) (fun (v : Set.{u1} E) => And (IsOpen.{u1} E _inst_4 u) (And (IsOpen.{u1} E _inst_4 v) (And (Convex.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) u) (And (Convex.{u2, u1} 𝕜 E (OrderedCommSemiring.toOrderedSemiring.{u2} 𝕜 (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (LinearOrderedSemifield.toSemifield.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_2))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (StrictOrderedSemiring.toSemiring.{u2} 𝕜 (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} 𝕜 (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} 𝕜 (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_2) _inst_3)))) v) (And (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s u) (And (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) t v) (Disjoint.{u1} (Set.{u1} E) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} E) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} E) (Preorder.toLE.{u1} (Set.{u1} E) (PartialOrder.toPreorder.{u1} (Set.{u1} E) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} E) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} E) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} E) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} E) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} E) (Set.instCompleteBooleanAlgebraSet.{u1} E)))))) u v))))))))) +Case conversion may be inaccurate. Consider using '#align disjoint.exists_open_convexes Disjoint.exists_open_convexesₓ'. -/ /-- In a locally convex space, if `s`, `t` are disjoint convex sets, `s` is compact and `t` is closed, then we can find open disjoint convex sets containing them. -/ theorem Disjoint.exists_open_convexes [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t) @@ -154,7 +208,13 @@ section LatticeOps variable {ι : Sort _} {𝕜 E F : Type _} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] -theorem locallyConvexSpaceInf {ts : Set (TopologicalSpace E)} +/- warning: locally_convex_space_Inf -> locallyConvexSpace_sInf is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] {ts : Set.{u2} (TopologicalSpace.{u2} E)}, (forall (t : TopologicalSpace.{u2} E), (Membership.Mem.{u2, u2} (TopologicalSpace.{u2} E) (Set.{u2} (TopologicalSpace.{u2} E)) (Set.hasMem.{u2} (TopologicalSpace.{u2} E)) t ts) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t)) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 (InfSet.sInf.{u2} (TopologicalSpace.{u2} E) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} E) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} E) (TopologicalSpace.completeLattice.{u2} E))) ts)) +but is expected to have type + forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] {ts : Set.{u2} (TopologicalSpace.{u2} E)}, (forall (t : TopologicalSpace.{u2} E), (Membership.mem.{u2, u2} (TopologicalSpace.{u2} E) (Set.{u2} (TopologicalSpace.{u2} E)) (Set.instMembershipSet.{u2} (TopologicalSpace.{u2} E)) t ts) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t)) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 (InfSet.sInf.{u2} (TopologicalSpace.{u2} E) (ConditionallyCompleteLattice.toInfSet.{u2} (TopologicalSpace.{u2} E) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} E) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} E))) ts)) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_Inf locallyConvexSpace_sInfₓ'. -/ +theorem locallyConvexSpace_sInf {ts : Set (TopologicalSpace E)} (h : ∀ t ∈ ts, @LocallyConvexSpace 𝕜 E _ _ _ t) : @LocallyConvexSpace 𝕜 E _ _ _ (sInf ts) := by letI : TopologicalSpace E := Inf ts @@ -165,29 +225,43 @@ theorem locallyConvexSpaceInf {ts : Set (TopologicalSpace E)} (fun x => _) fun x If hif => convex_iInter fun i => convex_iInter fun hi => (hif.2 i hi).2 rw [nhds_sInf, ← iInf_subtype''] exact has_basis_infi' fun i : ts => (@locallyConvexSpace_iff 𝕜 E _ _ _ ↑i).mp (h (↑i) i.2) x -#align locally_convex_space_Inf locallyConvexSpaceInf - -theorem locallyConvexSpaceInfi {ts' : ι → TopologicalSpace E} +#align locally_convex_space_Inf locallyConvexSpace_sInf + +/- warning: locally_convex_space_infi -> locallyConvexSpace_iInf is a dubious translation: +lean 3 declaration is + forall {ι : Sort.{u1}} {𝕜 : Type.{u2}} {E : Type.{u3}} [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u3} E] [_inst_3 : Module.{u2, u3} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] {ts' : ι -> (TopologicalSpace.{u3} E)}, (forall (i : ι), LocallyConvexSpace.{u2, u3} 𝕜 E _inst_1 _inst_2 _inst_3 (ts' i)) -> (LocallyConvexSpace.{u2, u3} 𝕜 E _inst_1 _inst_2 _inst_3 (iInf.{u3, u1} (TopologicalSpace.{u3} E) (ConditionallyCompleteLattice.toHasInf.{u3} (TopologicalSpace.{u3} E) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} E) (TopologicalSpace.completeLattice.{u3} E))) ι (fun (i : ι) => ts' i))) +but is expected to have type + forall {ι : Sort.{u1}} {𝕜 : Type.{u2}} {E : Type.{u3}} [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u3} E] [_inst_3 : Module.{u2, u3} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] {ts' : ι -> (TopologicalSpace.{u3} E)}, (forall (i : ι), LocallyConvexSpace.{u2, u3} 𝕜 E _inst_1 _inst_2 _inst_3 (ts' i)) -> (LocallyConvexSpace.{u2, u3} 𝕜 E _inst_1 _inst_2 _inst_3 (iInf.{u3, u1} (TopologicalSpace.{u3} E) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} E) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} E) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} E))) ι (fun (i : ι) => ts' i))) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_infi locallyConvexSpace_iInfₓ'. -/ +theorem locallyConvexSpace_iInf {ts' : ι → TopologicalSpace E} (h' : ∀ i, @LocallyConvexSpace 𝕜 E _ _ _ (ts' i)) : @LocallyConvexSpace 𝕜 E _ _ _ (⨅ i, ts' i) := by - refine' locallyConvexSpaceInf _ + refine' locallyConvexSpace_sInf _ rwa [forall_range_iff] -#align locally_convex_space_infi locallyConvexSpaceInfi - -/- warning: locally_convex_space_inf clashes with locally_convex_space_Inf -> locallyConvexSpaceInf -Case conversion may be inaccurate. Consider using '#align locally_convex_space_inf locallyConvexSpaceInfₓ'. -/ -#print locallyConvexSpaceInf /- -theorem locallyConvexSpaceInf {t₁ t₂ : TopologicalSpace E} (h₁ : @LocallyConvexSpace 𝕜 E _ _ _ t₁) +#align locally_convex_space_infi locallyConvexSpace_iInf + +/- warning: locally_convex_space_inf -> locallyConvexSpace_inf is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] {t₁ : TopologicalSpace.{u2} E} {t₂ : TopologicalSpace.{u2} E}, (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t₁) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t₂) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 (Inf.inf.{u2} (TopologicalSpace.{u2} E) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.{u2} E) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.{u2} E) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} E) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} E) (TopologicalSpace.completeLattice.{u2} E))))) t₁ t₂)) +but is expected to have type + forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] {t₁ : TopologicalSpace.{u2} E} {t₂ : TopologicalSpace.{u2} E}, (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t₁) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 t₂) -> (LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 (Inf.inf.{u2} (TopologicalSpace.{u2} E) (Lattice.toInf.{u2} (TopologicalSpace.{u2} E) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.{u2} E) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} E) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u2} E)))) t₁ t₂)) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_inf locallyConvexSpace_infₓ'. -/ +theorem locallyConvexSpace_inf {t₁ t₂ : TopologicalSpace E} (h₁ : @LocallyConvexSpace 𝕜 E _ _ _ t₁) (h₂ : @LocallyConvexSpace 𝕜 E _ _ _ t₂) : @LocallyConvexSpace 𝕜 E _ _ _ (t₁ ⊓ t₂) := by rw [inf_eq_iInf] - refine' locallyConvexSpaceInfi fun b => _ + refine' locallyConvexSpace_iInf fun b => _ cases b <;> assumption -#align locally_convex_space_inf locallyConvexSpaceInf --/ - -theorem locallyConvexSpaceInduced {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F] +#align locally_convex_space_inf locallyConvexSpace_inf + +/- warning: locally_convex_space_induced -> locallyConvexSpace_induced is a dubious translation: +lean 3 declaration is + forall {𝕜 : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : OrderedSemiring.{u1} 𝕜] [_inst_2 : AddCommMonoid.{u2} E] [_inst_3 : Module.{u1, u2} 𝕜 E (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2] [_inst_4 : AddCommMonoid.{u3} F] [_inst_5 : Module.{u1, u3} 𝕜 F (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_4] {t : TopologicalSpace.{u3} F} [_inst_6 : LocallyConvexSpace.{u1, u3} 𝕜 F _inst_1 _inst_4 _inst_5 t] (f : LinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))) E F _inst_2 _inst_4 _inst_3 _inst_5), LocallyConvexSpace.{u1, u2} 𝕜 E _inst_1 _inst_2 _inst_3 (TopologicalSpace.induced.{u2, u3} E F (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))) E F _inst_2 _inst_4 _inst_3 _inst_5) (fun (_x : LinearMap.{u1, u1, u2, u3} 𝕜 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1))) E F _inst_2 _inst_4 _inst_3 _inst_5) => E -> F) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} 𝕜 𝕜 E F (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1) _inst_2 _inst_4 _inst_3 _inst_5 (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (OrderedSemiring.toSemiring.{u1} 𝕜 _inst_1)))) f) t) +but is expected to have type + forall {𝕜 : Type.{u2}} {E : Type.{u1}} {F : Type.{u3}} [_inst_1 : OrderedSemiring.{u2} 𝕜] [_inst_2 : AddCommMonoid.{u1} E] [_inst_3 : Module.{u2, u1} 𝕜 E (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2] [_inst_4 : AddCommMonoid.{u3} F] [_inst_5 : Module.{u2, u3} 𝕜 F (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_4] {t : TopologicalSpace.{u3} F} [_inst_6 : LocallyConvexSpace.{u2, u3} 𝕜 F _inst_1 _inst_4 _inst_5 t] (f : LinearMap.{u2, u2, u1, u3} 𝕜 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) E F _inst_2 _inst_4 _inst_3 _inst_5), LocallyConvexSpace.{u2, u1} 𝕜 E _inst_1 _inst_2 _inst_3 (TopologicalSpace.induced.{u1, u3} E F (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (LinearMap.{u2, u2, u1, u3} 𝕜 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1))) E F _inst_2 _inst_4 _inst_3 _inst_5) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => F) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u3} 𝕜 𝕜 E F (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1) _inst_2 _inst_4 _inst_3 _inst_5 (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (OrderedSemiring.toSemiring.{u2} 𝕜 _inst_1)))) f) t) +Case conversion may be inaccurate. Consider using '#align locally_convex_space_induced locallyConvexSpace_inducedₓ'. -/ +theorem locallyConvexSpace_induced {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F] (f : E →ₗ[𝕜] F) : @LocallyConvexSpace 𝕜 E _ _ _ (t.induced f) := by letI : TopologicalSpace E := t.induced f @@ -197,16 +271,16 @@ theorem locallyConvexSpaceInduced {t : TopologicalSpace F} [LocallyConvexSpace hs.linear_preimage f rw [nhds_induced] exact (LocallyConvexSpace.convex_basis <| f x).comap f -#align locally_convex_space_induced locallyConvexSpaceInduced +#align locally_convex_space_induced locallyConvexSpace_induced instance {ι : Type _} {X : ι → Type _} [∀ i, AddCommMonoid (X i)] [∀ i, TopologicalSpace (X i)] [∀ i, Module 𝕜 (X i)] [∀ i, LocallyConvexSpace 𝕜 (X i)] : LocallyConvexSpace 𝕜 (∀ i, X i) := - locallyConvexSpaceInfi fun i => locallyConvexSpaceInduced (LinearMap.proj i) + locallyConvexSpace_iInf fun i => locallyConvexSpace_induced (LinearMap.proj i) instance [TopologicalSpace E] [TopologicalSpace F] [LocallyConvexSpace 𝕜 E] [LocallyConvexSpace 𝕜 F] : LocallyConvexSpace 𝕜 (E × F) := - locallyConvexSpaceInf (locallyConvexSpaceInduced (LinearMap.fst _ _ _)) - (locallyConvexSpaceInduced (LinearMap.snd _ _ _)) + locallyConvexSpace_inf (locallyConvexSpace_induced (LinearMap.fst _ _ _)) + (locallyConvexSpace_induced (LinearMap.snd _ _ _)) end LatticeOps diff --git a/lake-manifest.json b/lake-manifest.json index 2c60ea6b37..4b5b013ea2 100644 --- a/lake-manifest.json +++ b/lake-manifest.json @@ -4,15 +4,15 @@ [{"git": {"url": "https://github.com/leanprover-community/lean3port.git", "subDir?": null, - "rev": "882a5a2b0b8f43ca4dc9b89a5447565e6c9e9196", + "rev": "cbdbb3d294a4e34fb4d30f2a134a8700880238a9", "name": "lean3port", - "inputRev?": "882a5a2b0b8f43ca4dc9b89a5447565e6c9e9196"}}, + "inputRev?": "cbdbb3d294a4e34fb4d30f2a134a8700880238a9"}}, {"git": {"url": "https://github.com/leanprover-community/mathlib4.git", "subDir?": null, - "rev": "54991d4b205a7765d2f9107dd410f3c7b793da2b", + "rev": "e7aad4d80fed7693efcb331dcf91f09c9b51d3ea", "name": "mathlib", - "inputRev?": "54991d4b205a7765d2f9107dd410f3c7b793da2b"}}, + "inputRev?": "e7aad4d80fed7693efcb331dcf91f09c9b51d3ea"}}, {"git": {"url": "https://github.com/gebner/quote4", "subDir?": null, diff --git a/lakefile.lean b/lakefile.lean index 3f74430117..e7cc2e8431 100644 --- a/lakefile.lean +++ b/lakefile.lean @@ -4,7 +4,7 @@ open Lake DSL System -- Usually the `tag` will be of the form `nightly-2021-11-22`. -- If you would like to use an artifact from a PR build, -- it will be of the form `pr-branchname-sha`. -def tag : String := "nightly-2023-05-21-16" +def tag : String := "nightly-2023-05-21-18" def releaseRepo : String := "leanprover-community/mathport" def oleanTarName : String := "mathlib3-binport.tar.gz" @@ -38,7 +38,7 @@ target fetchOleans (_pkg : Package) : Unit := do untarReleaseArtifact releaseRepo tag oleanTarName libDir return .nil -require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"882a5a2b0b8f43ca4dc9b89a5447565e6c9e9196" +require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"cbdbb3d294a4e34fb4d30f2a134a8700880238a9" @[default_target] lean_lib Mathbin where