| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| open scoped MeasureTheory |
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| namespace Problem367 |
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| theorem condExp_comp_measurePreserving_equiv |
| {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {mSmall m : MeasurableSpace Ω} {μ : Measure Ω} [IsFiniteMeasure μ] |
| (hm : mSmall ≤ m) |
| (e : Ω ≃ᵐ Ω) (he : MeasurePreserving e μ μ) |
| (heSmall : @Measurable Ω Ω mSmall mSmall e.toEquiv) |
| (heSymmSmall : @Measurable Ω Ω mSmall mSmall e.toEquiv.symm) |
| {f : Ω → E} (hf : Integrable f μ) : |
| (μ[f | mSmall]) ∘ e =ᵐ[μ] μ[f ∘ e | mSmall] := by |
| have hfc : Integrable (f ∘ e) μ := he.integrable_comp_of_integrable hf |
| have hgc : Integrable ((μ[f | mSmall]) ∘ e) μ := |
| he.integrable_comp_of_integrable integrable_condExp |
| apply ae_eq_condExp_of_forall_setIntegral_eq hm hfc (fun s _ _ => hgc.integrableOn) |
| · intro s hs _ |
| have hImage : MeasurableSet[mSmall] (e '' s) := by |
| rw [e.image_eq_preimage_symm] |
| exact heSymmSmall hs |
| calc |
| (∫ x in s, ((μ[f | mSmall]) ∘ e) x ∂μ) = |
| ∫ x in e '' s, (μ[f | mSmall]) x ∂μ := |
| (he.setIntegral_image_emb e.measurableEmbedding _ _).symm |
| _ = ∫ x in e '' s, f x ∂μ := setIntegral_condExp hm hf hImage |
| _ = ∫ x in s, (f ∘ e) x ∂μ := |
| he.setIntegral_image_emb e.measurableEmbedding _ _ |
| · exact (stronglyMeasurable_condExp.comp_measurable heSmall).aestronglyMeasurable |
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| theorem condExp_ae_invariant_of_ae_invariant |
| {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {mSmall m : MeasurableSpace Ω} {μ : Measure Ω} [IsFiniteMeasure μ] |
| (hm : mSmall ≤ m) |
| (e : Ω ≃ᵐ Ω) (he : MeasurePreserving e μ μ) |
| (heSmall : @Measurable Ω Ω mSmall mSmall e.toEquiv) |
| (heSymmSmall : @Measurable Ω Ω mSmall mSmall e.toEquiv.symm) |
| {f : Ω → E} (hf : Integrable f μ) (hinv : f ∘ e =ᵐ[μ] f) : |
| (μ[f | mSmall]) ∘ e =ᵐ[μ] μ[f | mSmall] := |
| (condExp_comp_measurePreserving_equiv hm e he heSmall heSymmSmall hf).trans |
| (condExp_congr_ae hinv) |
|
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| def finiteGroupAverage |
| {G Ω E : Type*} [Group G] [Fintype G] [MulAction G Ω] |
| [AddCommGroup E] [Module ℝ E] (f : Ω → E) (x : Ω) : E := |
| (Fintype.card G : ℝ)⁻¹ • ∑ g : G, f (g • x) |
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| theorem finiteGroupAverage_invariant |
| {G Ω E : Type*} [Group G] [Fintype G] [MulAction G Ω] |
| [AddCommGroup E] [Module ℝ E] (f : Ω → E) (h : G) (x : Ω) : |
| finiteGroupAverage (G := G) f (h • x) = finiteGroupAverage (G := G) f x := by |
| unfold finiteGroupAverage |
| congr 1 |
| simp_rw [← mul_smul] |
| exact Equiv.sum_comp (Equiv.mulRight h) (fun g => f (g • x)) |
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| theorem stronglyMeasurable_finiteGroupAverage |
| {G Ω E : Type*} [Group G] [Fintype G] [MulAction G Ω] |
| [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (hact : ∀ g : G, Measurable (fun x : Ω => g • x)) |
| {f : Ω → E} (hf : StronglyMeasurable f) : |
| StronglyMeasurable (finiteGroupAverage (G := G) f) := by |
| exact (Finset.stronglyMeasurable_fun_sum Finset.univ |
| (fun g _ => hf.comp_measurable (hact g))).fun_const_smul ((Fintype.card G : ℝ)⁻¹) |
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| theorem finiteGroupAverage_ae_eq |
| {G Ω E : Type*} [Group G] [Fintype G] [MulAction G Ω] |
| [MeasurableSpace Ω] [AddCommGroup E] [Module ℝ E] |
| {μ : Measure Ω} {f : Ω → E} |
| (hinv : ∀ g : G, (fun x => f (g • x)) =ᵐ[μ] f) : |
| finiteGroupAverage (G := G) f =ᵐ[μ] f := by |
| have hall : ∀ᵐ x ∂μ, ∀ g : G, f (g • x) = f x := ae_all_iff.mpr hinv |
| filter_upwards [hall] with x hx |
| simp only [finiteGroupAverage, hx, Finset.sum_const, Finset.card_univ, ← Nat.cast_smul_eq_nsmul ℝ, smul_smul] |
| simp |
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| theorem exists_stronglyMeasurable_invariant_version |
| {G Ω E : Type*} [Group G] [Fintype G] [MulAction G Ω] |
| [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {μ : Measure Ω} (hact : ∀ g : G, Measurable (fun x : Ω => g • x)) |
| {f : Ω → E} (hf : StronglyMeasurable f) |
| (hinv : ∀ g : G, (fun x => f (g • x)) =ᵐ[μ] f) : |
| ∃ f' : Ω → E, StronglyMeasurable f' ∧ f' =ᵐ[μ] f ∧ |
| ∀ (g : G) (x : Ω), f' (g • x) = f' x := |
| ⟨finiteGroupAverage (G := G) f, stronglyMeasurable_finiteGroupAverage hact hf, |
| finiteGroupAverage_ae_eq hinv, finiteGroupAverage_invariant f⟩ |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
