Skip to content

Commit

Permalink
bump to nightly-2023-05-06-08
Browse files Browse the repository at this point in the history
  • Loading branch information
leanprover-community-bot committed May 6, 2023
1 parent bfb6ebc commit e8904d9
Show file tree
Hide file tree
Showing 11 changed files with 281 additions and 147 deletions.
2 changes: 1 addition & 1 deletion Mathbin/MeasureTheory/Group/AddCircle.lean
Expand Up @@ -79,7 +79,7 @@ theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCir
· simpa only [Ne.def, AddSubgroup.mk_eq_zero_iff] using hg'
change ae_disjoint volume (g +ᵥ I) I
refine'
ae_disjoint.congr (Disjoint.aeDisjoint _)
ae_disjoint.congr (Disjoint.aedisjoint _)
((quasi_measure_preserving_add_left volume (-g)).vadd_ae_eq_of_ae_eq g hI) hI
have hBg : g +ᵥ B = ball (g + x) (T / (2 * n)) := by
rw [add_comm g x, ← singleton_add_ball _ x g, add_ball, thickening_singleton]
Expand Down
40 changes: 20 additions & 20 deletions Mathbin/MeasureTheory/Group/FundamentalDomain.lean
Expand Up @@ -55,7 +55,7 @@ structure IsAddFundamentalDomain (G : Type _) {α : Type _} [Zero G] [VAdd G α]
(s : Set α) (μ : Measure α := by exact MeasureTheory.MeasureSpace.volume) : Prop where
NullMeasurableSet : NullMeasurableSet s μ
ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s
AeDisjoint : Pairwise <| (AeDisjoint μ on fun g : G => g +ᵥ s)
AEDisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)
#align measure_theory.is_add_fundamental_domain MeasureTheory.IsAddFundamentalDomain

/-- A measurable set `s` is a *fundamental domain* for an action of a group `G` on a measurable
Expand All @@ -66,7 +66,7 @@ structure IsFundamentalDomain (G : Type _) {α : Type _} [One G] [SMul G α] [Me
(s : Set α) (μ : Measure α := by exact MeasureTheory.MeasureSpace.volume) : Prop where
NullMeasurableSet : NullMeasurableSet s μ
ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s
AeDisjoint : Pairwise <| (AeDisjoint μ on fun g : G => g • s)
AEDisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)
#align measure_theory.is_fundamental_domain MeasureTheory.IsFundamentalDomain
#align measure_theory.is_add_fundamental_domain MeasureTheory.IsAddFundamentalDomain

Expand All @@ -85,8 +85,8 @@ theorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G
IsFundamentalDomain G s μ :=
{ NullMeasurableSet := h_meas
ae_covers := eventually_of_forall fun x => (h_exists x).exists
AeDisjoint := fun a b hab =>
Disjoint.aeDisjoint <|
AEDisjoint := fun a b hab =>
Disjoint.aedisjoint <|
disjoint_left.2 fun x hxa hxb =>
by
rw [mem_smul_set_iff_inv_smul_mem] at hxa hxb
Expand All @@ -99,11 +99,11 @@ theorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G
@[to_additive
"For `s` to be a fundamental domain, it's enough to check `ae_disjoint (g +ᵥ s) s` for\n`g ≠ 0`."]
theorem mk'' (h_meas : NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s)
(h_ae_disjoint : ∀ (g) (_ : g ≠ (1 : G)), AeDisjoint μ (g • s) s)
(h_ae_disjoint : ∀ (g) (_ : g ≠ (1 : G)), AEDisjoint μ (g • s) s)
(h_qmp : ∀ g : G, QuasiMeasurePreserving ((· • ·) g : α → α) μ μ) : IsFundamentalDomain G s μ :=
{ NullMeasurableSet := h_meas
ae_covers := h_ae_covers
AeDisjoint := pairwise_aeDisjoint_of_aeDisjoint_forall_ne_one h_ae_disjoint h_qmp }
AEDisjoint := pairwise_aEDisjoint_of_aEDisjoint_forall_ne_one h_ae_disjoint h_qmp }
#align measure_theory.is_fundamental_domain.mk'' MeasureTheory.IsFundamentalDomain.mk''
#align measure_theory.is_add_fundamental_domain.mk'' MeasureTheory.IsAddFundamentalDomain.mk''

Expand All @@ -115,13 +115,13 @@ sufficiently large. -/
@[to_additive MeasureTheory.IsAddFundamentalDomain.mk_of_measure_univ_le
"\nIf a measurable space has a finite measure `μ` and a countable additive group `G` acts\nquasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient\nto check that its translates `g +ᵥ s` are (almost) disjoint and that the sum `∑' g, μ (g +ᵥ s)` is\nsufficiently large."]
theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
(h_ae_disjoint : ∀ (g) (_ : g ≠ (1 : G)), AeDisjoint μ (g • s) s)
(h_ae_disjoint : ∀ (g) (_ : g ≠ (1 : G)), AEDisjoint μ (g • s) s)
(h_qmp : ∀ g : G, QuasiMeasurePreserving ((· • ·) g : α → α) μ μ)
(h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=
have ae_disjoint : Pairwise (AeDisjoint μ on fun g : G => g • s) :=
pairwise_aeDisjoint_of_aeDisjoint_forall_ne_one h_ae_disjoint h_qmp
have ae_disjoint : Pairwise (AEDisjoint μ on fun g : G => g • s) :=
pairwise_aEDisjoint_of_aEDisjoint_forall_ne_one h_ae_disjoint h_qmp
{ NullMeasurableSet := h_meas
AeDisjoint
AEDisjoint
ae_covers :=
by
replace h_meas : ∀ g : G, null_measurable_set (g • s) μ := fun g =>
Expand Down Expand Up @@ -149,7 +149,7 @@ theorem unionᵢ_smul_ae_eq (h : IsFundamentalDomain G s μ) : (⋃ g : G, g •
@[to_additive]
theorem mono (h : IsFundamentalDomain G s μ) {ν : Measure α} (hle : ν ≪ μ) :
IsFundamentalDomain G s ν :=
⟨h.1.mono_ac hle, hle h.2, h.AeDisjoint.mono fun a b hab => hle hab⟩
⟨h.1.mono_ac hle, hle h.2, h.AEDisjoint.mono fun a b hab => hle hab⟩
#align measure_theory.is_fundamental_domain.mono MeasureTheory.IsFundamentalDomain.mono
#align measure_theory.is_add_fundamental_domain.mono MeasureTheory.IsAddFundamentalDomain.mono

Expand All @@ -159,7 +159,7 @@ theorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f
(hef : ∀ g, Semiconj f ((· • ·) (e g)) ((· • ·) g)) : IsFundamentalDomain H (f ⁻¹' s) ν :=
{ NullMeasurableSet := h.NullMeasurableSet.Preimage hf
ae_covers := (hf.ae h.ae_covers).mono fun x ⟨g, hg⟩ => ⟨e g, by rwa [mem_preimage, hef g x]⟩
AeDisjoint := fun a b hab => by
AEDisjoint := fun a b hab => by
lift e to G ≃ H using he
have : (e.symm a⁻¹)⁻¹ ≠ (e.symm b⁻¹)⁻¹ := by simp [hab]
convert(h.ae_disjoint this).Preimage hf using 1
Expand All @@ -180,11 +180,11 @@ theorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f :
#align measure_theory.is_add_fundamental_domain.image_of_equiv MeasureTheory.IsAddFundamentalDomain.image_of_equiv

@[to_additive]
theorem pairwise_aeDisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
Pairwise fun g₁ g₂ : G => AeDisjoint ν (g₁ • s) (g₂ • s) :=
h.AeDisjoint.mono fun g₁ g₂ H => hν H
#align measure_theory.is_fundamental_domain.pairwise_ae_disjoint_of_ac MeasureTheory.IsFundamentalDomain.pairwise_aeDisjoint_of_ac
#align measure_theory.is_add_fundamental_domain.pairwise_ae_disjoint_of_ac MeasureTheory.IsAddFundamentalDomain.pairwise_aeDisjoint_of_ac
theorem pairwise_aEDisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
Pairwise fun g₁ g₂ : G => AEDisjoint ν (g₁ • s) (g₂ • s) :=
h.AEDisjoint.mono fun g₁ g₂ H => hν H
#align measure_theory.is_fundamental_domain.pairwise_ae_disjoint_of_ac MeasureTheory.IsFundamentalDomain.pairwise_aEDisjoint_of_ac
#align measure_theory.is_add_fundamental_domain.pairwise_ae_disjoint_of_ac MeasureTheory.IsAddFundamentalDomain.pairwise_aEDisjoint_of_ac

@[to_additive]
theorem smul_of_comm {G' : Type _} [Group G'] [MulAction G' α] [MeasurableSpace G']
Expand Down Expand Up @@ -505,7 +505,7 @@ has measure at most `μ s`. -/
@[to_additive
"If the additive action of a countable group `G` admits an invariant measure `μ` with\na fundamental domain `s`, then every null-measurable set `t` such that the sets `g +ᵥ t ∩ s` are\npairwise a.e.-disjoint has measure at most `μ s`."]
theorem measure_le_of_pairwise_disjoint (hs : IsFundamentalDomain G s μ)
(ht : NullMeasurableSet t μ) (hd : Pairwise (AeDisjoint μ on fun g : G => g • t ∩ s)) :
(ht : NullMeasurableSet t μ) (hd : Pairwise (AEDisjoint μ on fun g : G => g • t ∩ s)) :
μ t ≤ μ s :=
calc
μ t = ∑' g : G, μ (g • t ∩ s) := hs.measure_eq_tsum t
Expand All @@ -527,7 +527,7 @@ theorem exists_ne_one_smul_eq (hs : IsFundamentalDomain G s μ) (htm : NullMeasu
(ht : μ s < μ t) : ∃ (x : _)(_ : x ∈ t)(y : _)(_ : y ∈ t)(g : _)(_ : g ≠ (1 : G)), g • x = y :=
by
contrapose! ht
refine' hs.measure_le_of_pairwise_disjoint htm (Pairwise.aeDisjoint fun g₁ g₂ hne => _)
refine' hs.measure_le_of_pairwise_disjoint htm (Pairwise.aedisjoint fun g₁ g₂ hne => _)
dsimp [Function.onFun]
refine' (Disjoint.inf_left _ _).inf_right _
rw [Set.disjoint_left]
Expand Down Expand Up @@ -743,7 +743,7 @@ protected theorem fundamentalInterior : IsFundamentalDomain G (fundamentalInteri
measure_union_null
(measure_Union_null fun _ => measure_smul_null hs.measure_fundamental_frontier _)
hs.ae_covers
AeDisjoint := (pairwise_disjoint_fundamentalInterior _ _).mono fun _ _ => Disjoint.aeDisjoint }
AEDisjoint := (pairwise_disjoint_fundamentalInterior _ _).mono fun _ _ => Disjoint.aedisjoint }
#align measure_theory.is_fundamental_domain.fundamental_interior MeasureTheory.IsFundamentalDomain.fundamentalInterior

end IsFundamentalDomain
Expand Down
2 changes: 1 addition & 1 deletion Mathbin/MeasureTheory/Group/GeometryOfNumbers.lean
Expand Up @@ -59,7 +59,7 @@ theorem exists_pair_mem_lattice_not_disjoint_vadd [AddCommGroup L] [Countable L]
exact
((fund.measure_eq_tsum _).trans
(measure_Union₀
(Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).AeDisjoint)
(Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).AEDisjoint)
fun _ => (hS.vadd _).inter fund.null_measurable_set).symm).trans_le
(measure_mono <| Union_subset fun _ => inter_subset_right _ _)
#align measure_theory.exists_pair_mem_lattice_not_disjoint_vadd MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd
Expand Down
6 changes: 3 additions & 3 deletions Mathbin/MeasureTheory/Integral/Average.lean
Expand Up @@ -157,7 +157,7 @@ theorem measure_smul_set_average (f : α → E) {s : Set α} (h : μ s ≠ ∞)
rw [← measure_smul_average, restrict_apply_univ]
#align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_set_average

theorem average_union {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t) (ht : NullMeasurableSet t μ)
theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
(hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
(⨍ x in s ∪ t, f x ∂μ) =
(((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ) +
Expand All @@ -167,7 +167,7 @@ theorem average_union {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t) (ht
rw [restrict_union₀ hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]
#align measure_theory.average_union MeasureTheory.average_union

theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t)
theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
(ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞)
(hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
(⨍ x in s ∪ t, f x ∂μ) ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) :=
Expand All @@ -179,7 +179,7 @@ theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AeDisj
⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩
#align measure_theory.average_union_mem_open_segment MeasureTheory.average_union_mem_openSegment

theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AeDisjoint μ s t)
theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t)
(ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
(hft : IntegrableOn f t μ) :
(⨍ x in s ∪ t, f x ∂μ) ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] :=
Expand Down
12 changes: 6 additions & 6 deletions Mathbin/MeasureTheory/Integral/Lebesgue.lean
Expand Up @@ -1216,19 +1216,19 @@ 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)
(hd : Pairwise (AeDisjoint μ on s)) (f : α → ℝ≥0∞) :
(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

theorem lintegral_unionᵢ [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_Union₀ (fun i => (hm i).NullMeasurableSet) hd.AEDisjoint f
#align measure_theory.lintegral_Union MeasureTheory.lintegral_unionᵢ

theorem lintegral_bUnion₀ {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∞) :
(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
Expand All @@ -1238,19 +1238,19 @@ theorem lintegral_bUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable)
theorem lintegral_bUnion {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
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}
(hd : Set.Pairwise (↑s) (AeDisjoint μ on t)) (hm : ∀ b ∈ s, NullMeasurableSet (t b) μ)
(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)
(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
lintegral_bUnion_finset₀ hd.AEDisjoint (fun b hb => (hm b hb).NullMeasurableSet) f
#align measure_theory.lintegral_bUnion_finset MeasureTheory.lintegral_bUnion_finset

theorem lintegral_unionᵢ_le [Countable β] (s : β → Set α) (f : α → ℝ≥0∞) :
Expand Down
12 changes: 6 additions & 6 deletions Mathbin/MeasureTheory/Integral/SetIntegral.lean
Expand Up @@ -94,15 +94,15 @@ theorem set_integral_congr_set_ae (hst : s =ᵐ[μ] t) : (∫ x in s, f x ∂μ)
rw [measure.restrict_congr_set hst]
#align measure_theory.set_integral_congr_set_ae MeasureTheory.set_integral_congr_set_ae

theorem integral_union_ae (hst : AeDisjoint μ s t) (ht : NullMeasurableSet t μ)
theorem integral_union_ae (hst : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
(hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
(∫ x in s ∪ t, f x ∂μ) = (∫ x in s, f x ∂μ) + ∫ x in t, f x ∂μ := by
simp only [integrable_on, measure.restrict_union₀ hst ht, integral_add_measure hfs hft]
#align measure_theory.integral_union_ae MeasureTheory.integral_union_ae

theorem integral_union (hst : Disjoint s t) (ht : MeasurableSet t) (hfs : IntegrableOn f s μ)
(hft : IntegrableOn f t μ) : (∫ x in s ∪ t, f x ∂μ) = (∫ x in s, f x ∂μ) + ∫ x in t, f x ∂μ :=
integral_union_ae hst.AeDisjoint ht.NullMeasurableSet hfs hft
integral_union_ae hst.AEDisjoint ht.NullMeasurableSet hfs hft
#align measure_theory.integral_union MeasureTheory.integral_union

theorem integral_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) (hts : t ⊆ s) :
Expand Down Expand Up @@ -160,7 +160,7 @@ theorem integral_univ : (∫ x in univ, f x ∂μ) = ∫ x, f x ∂μ := by rw [
theorem integral_add_compl₀ (hs : NullMeasurableSet s μ) (hfi : Integrable f μ) :
((∫ x in s, f x ∂μ) + ∫ x in sᶜ, f x ∂μ) = ∫ x, f x ∂μ := by
rw [←
integral_union_ae (@disjoint_compl_right (Set α) _ _).AeDisjoint hs.compl hfi.integrable_on
integral_union_ae (@disjoint_compl_right (Set α) _ _).AEDisjoint hs.compl hfi.integrable_on
hfi.integrable_on,
union_compl_self, integral_univ]
#align measure_theory.integral_add_compl₀ MeasureTheory.integral_add_compl₀
Expand Down Expand Up @@ -243,7 +243,7 @@ theorem tendsto_set_integral_of_monotone {ι : Type _} [Countable ι] [Semilatti
#align measure_theory.tendsto_set_integral_of_monotone MeasureTheory.tendsto_set_integral_of_monotone

theorem hasSum_integral_unionᵢ_ae {ι : Type _} [Countable ι] {s : ι → Set α}
(hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AeDisjoint μ on s))
(hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
(hfi : IntegrableOn f (⋃ i, s i) μ) :
HasSum (fun n => ∫ a in s n, f a ∂μ) (∫ a in ⋃ n, s n, f a ∂μ) :=
by
Expand All @@ -255,7 +255,7 @@ theorem hasSum_integral_unionᵢ {ι : Type _} [Countable ι] {s : ι → Set α
(hm : ∀ i, MeasurableSet (s i)) (hd : Pairwise (Disjoint on s))
(hfi : IntegrableOn f (⋃ i, s i) μ) :
HasSum (fun n => ∫ a in s n, f a ∂μ) (∫ a in ⋃ n, s n, f a ∂μ) :=
hasSum_integral_unionᵢ_ae (fun i => (hm i).NullMeasurableSet) (hd.mono fun i j h => h.AeDisjoint)
hasSum_integral_unionᵢ_ae (fun i => (hm i).NullMeasurableSet) (hd.mono fun i j h => h.AEDisjoint)
hfi
#align measure_theory.has_sum_integral_Union MeasureTheory.hasSum_integral_unionᵢ

Expand All @@ -266,7 +266,7 @@ theorem integral_unionᵢ {ι : Type _} [Countable ι] {s : ι → Set α} (hm :
#align measure_theory.integral_Union MeasureTheory.integral_unionᵢ

theorem integral_unionᵢ_ae {ι : Type _} [Countable ι] {s : ι → Set α}
(hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AeDisjoint μ on s))
(hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
(hfi : IntegrableOn f (⋃ i, s i) μ) : (∫ a in ⋃ n, s n, f a ∂μ) = ∑' n, ∫ a in s n, f a ∂μ :=
(HasSum.tsum_eq (hasSum_integral_unionᵢ_ae hm hd hfi)).symm
#align measure_theory.integral_Union_ae MeasureTheory.integral_unionᵢ_ae
Expand Down

0 comments on commit e8904d9

Please sign in to comment.