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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.Probability.FreeIsing.Conditioning.Pullback
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem iCondIndepFun_comp_measurePreserving
{Ω Ω' ι : Type*} {C : ι → Type*}
[mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω']
[mC : ∀ i, MeasurableSpace (C i)]
[StandardBorelSpace Ω] [StandardBorelSpace Ω']
(μ : Measure Ω) (ν : Measure Ω') [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(e : Ω → Ω') (he : MeasurePreserving e μ ν)
(m : MeasurableSpace Ω') (hm : m ≤ mΩ')
{f : ∀ i, Ω' → C i} (hf : ∀ i, @Measurable Ω' (C i) mΩ' (mC i) (f i))
(hfamily : iCondIndepFun m hm f ν) :
iCondIndepFun (m.comap e)
((MeasurableSpace.comap_mono hm).trans (@Measurable.comap_le Ω Ω' mΩ mΩ' e
(@MeasurePreserving.measurable Ω Ω' mΩ mΩ' e μ ν he)))
(fun i => f i ∘ e) μ := by
let : MeasurableSpace Ω' := mΩ'
classical
rw [iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC (fun i => f i ∘ e)
(fun i => (hf i).comp he.measurable)]
intro S sets hsets
let E : Set Ω' := ⋂ i ∈ S, f i ⁻¹' sets i
have hE : MeasurableSet E :=
S.measurableSet_biInter (fun i hi => (hsets i hi).preimage (hf i))
have hCE := condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator hE)
have heach : ∀ᵐ ω ∂μ, ∀ i ∈ S,
μ[((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ))) ∘ e | m.comap e] ω =
ν[(f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)) | m] (e ω) := by
apply S.eventually_all.mpr
intro i hi
exact condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator ((hsets i hi).preimage (hf i)))
have hmul := he.quasiMeasurePreserving.ae_eq_comp
((iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC f hf).mp hfamily S hsets)
filter_upwards [hCE, heach, hmul] with ω hωE hωeach hωmul
simp only [Finset.prod_apply, Function.comp_apply] at hωmul ⊢
have hindicator : (⋂ i ∈ S, (f i ∘ e) ⁻¹' sets i).indicator
(fun _ => (1 : ℝ)) = (E.indicator (fun _ => (1 : ℝ))) ∘ e := by
funext x
simp [E, Set.indicator, Function.comp_def]
rw [hindicator]
exact hωE.trans (hωmul.trans (Finset.prod_congr rfl (fun i hi => (hωeach i hi).symm)))
theorem iCondIndepFun_iff_comp_measurePreserving
{Ω Ω' ι : Type*} {C : ι → Type*}
[mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω']
[mC : ∀ i, MeasurableSpace (C i)]
[StandardBorelSpace Ω] [StandardBorelSpace Ω']
(μ : Measure Ω) (ν : Measure Ω') [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(e : Ω → Ω') (he : MeasurePreserving e μ ν)
(m : MeasurableSpace Ω') (hm : m ≤ mΩ')
{f : ∀ i, Ω' → C i} (hf : ∀ i, @Measurable Ω' (C i) mΩ' (mC i) (f i)) :
iCondIndepFun m hm f ν ↔
iCondIndepFun (m.comap e)
((MeasurableSpace.comap_mono hm).trans (@Measurable.comap_le Ω Ω' mΩ mΩ' e
(@MeasurePreserving.measurable Ω Ω' mΩ mΩ' e μ ν he)))
(fun i => f i ∘ e) μ := by
let : MeasurableSpace Ω' := mΩ'
classical
refine ⟨iCondIndepFun_comp_measurePreserving μ ν e he m hm hf, ?_⟩
intro hsource
rw [iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC f hf]
intro S sets hsets
have hmCE (h : Ω' → ℝ) : Measurable (ν[h | m]) :=
(stronglyMeasurable_condExp.mono hm).measurable
apply (ae_eq_iff_comp_measurePreserving μ ν e he (hmCE _) ?_).mpr
swap
· convert S.measurable_prod (fun i hi =>
hmCE ((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)))) using 1
ext ω'
simp only [Finset.prod_apply]
let E : Set Ω' := ⋂ i ∈ S, f i ⁻¹' sets i
have hE : MeasurableSet E :=
S.measurableSet_biInter (fun i hi => (hsets i hi).preimage (hf i))
have hCE := condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator hE)
have heach : ∀ᵐ ω ∂μ, ∀ i ∈ S,
μ[((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ))) ∘ e | m.comap e] ω =
ν[(f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)) | m] (e ω) := by
apply S.eventually_all.mpr
intro i hi
exact condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator ((hsets i hi).preimage (hf i)))
have hmul := (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC
(fun i => f i ∘ e) (fun i => (hf i).comp he.measurable)).mp hsource S hsets
filter_upwards [hCE, heach, hmul] with ω hωE hωeach hωmul
have hindicator : (⋂ i ∈ S, (f i ∘ e) ⁻¹' sets i).indicator
(fun _ => (1 : ℝ)) = (E.indicator (fun _ => (1 : ℝ))) ∘ e := by
funext x
simp [E, Set.indicator, Function.comp_def]
rw [hindicator] at hωmul
simp only [Function.comp_apply, Finset.prod_apply] at hωmul ⊢
exact hωE.symm.trans (hωmul.trans (Finset.prod_congr rfl (fun i hi => hωeach i hi)))
alias ⟨_, iCondIndepFun_of_comp_measurePreserving⟩ :=
iCondIndepFun_iff_comp_measurePreserving
end Problem367
end
end OAI
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