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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
open scoped MeasureTheory
namespace Problem367
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
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)
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)
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))
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)
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
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
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