[Merged by Bors] - feat: a function with vanishing integral against smooth functions supported in U is ae zero in U #8805
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| intro g g_smth g_supp | ||
| classical | ||
| have cpt := g_supp.image continuous_induced_dom | ||
| let g' x := if hx : x ∈ U then g ⟨x, hx⟩ else 0 |
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Could you instead define g' as Function.extend Subtype.val g 0, and prove in separate lemmas that the extension of a smooth compactly supported function is smooth and compactly supported?
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| theorem liftPropAt_iff_comp_subtype_val (hG : LocalInvariantProp G G' P) {U : Opens M} | ||
| (f : M → M') (x : U) : | ||
| LiftPropAt P f x ↔ LiftPropAt P (f ∘ Subtype.val) x := by | ||
| congrm ?_ ∧ ?_ | ||
| · simp_rw [continuousWithinAt_univ, U.openEmbedding'.continuousAt_iff] | ||
| · apply hG.congr_iff | ||
| exact (U.chartAt_subtype_val_symm_eventuallyEq).fun_comp (chartAt H' (f x) ∘ f) | ||
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| theorem liftPropAt_iff_comp_inclusion (hG : LocalInvariantProp G G' P) {U V : Opens M} (hUV : U ≤ V) | ||
| (f : V → M') (x : U) : | ||
| LiftPropAt P f (Set.inclusion hUV x) ↔ LiftPropAt P (f ∘ Set.inclusion hUV : U → M') x := by | ||
| congrm ?_ ∧ ?_ | ||
| · simp [continuousWithinAt_univ, | ||
| · simp_rw [continuousWithinAt_univ, | ||
| (TopologicalSpace.Opens.openEmbedding_of_le hUV).continuousAt_iff] | ||
| · apply hG.congr_iff | ||
| exact (TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq hUV).fun_comp | ||
| (chartAt H' (f (Set.inclusion hUV x)) ∘ f) | ||
| #align structure_groupoid.local_invariant_prop.lift_prop_at_iff_comp_inclusion StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_inclusion | ||
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| theorem liftProp_subtype_val {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q) | ||
| (hQ : ∀ y, Q id univ y) (U : Opens M) : | ||
| LiftProp Q (Subtype.val : U → M) := by | ||
| intro x | ||
| show LiftPropAt Q (id ∘ Subtype.val) x | ||
| rw [← hG.liftPropAt_iff_comp_subtype_val] | ||
| apply hG.liftProp_id hQ |
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cc author of #19416 @hrmacbeth in case I'm missing some easy way to deduce smooth_subtype_iff.
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…ported in U is ae zero in U (#8805) A stronger version of #8800, the differences are: + assume either `IsSigmaCompact U` or `SigmaCompactSpace M`; + only need test functions satisfying `tsupport g ⊆ U` rather than `support g ⊆ U`; + requires `LocallyIntegrableOn` U rather than `LocallyIntegrable` on the whole space. Also fills in some missing APIs around the manifold and measure theory libraries. Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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…ported in U is ae zero in U (#8805) A stronger version of #8800, the differences are: + assume either `IsSigmaCompact U` or `SigmaCompactSpace M`; + only need test functions satisfying `tsupport g ⊆ U` rather than `support g ⊆ U`; + requires `LocallyIntegrableOn` U rather than `LocallyIntegrable` on the whole space. Also fills in some missing APIs around the manifold and measure theory libraries. Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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A stronger version of #8800, the differences are:
assume either
IsSigmaCompact UorSigmaCompactSpace M;only need test functions satisfying
tsupport g ⊆ Urather thansupport g ⊆ U;requires
LocallyIntegrableOnU rather thanLocallyIntegrableon the whole space.Also fills in some missing APIs around the manifold and measure theory libraries.