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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 ProbabilityTheory
namespace Problem367
theorem setIntegral_comp_measurePreserving
{Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']
(μ : Measure Ω) (ν : Measure Ω') (e : Ω → Ω')
(he : MeasurePreserving e μ ν) {f : Ω' → ℝ}
(hf : AEStronglyMeasurable f ν) {B : Set Ω'} (hB : MeasurableSet B) :
(∫ ω in e ⁻¹' B, f (e ω) ∂μ) = ∫ ω' in B, f ω' ∂ν := by
have hres := he.restrict_preimage hB
have hfm : AEStronglyMeasurable f (Measure.map e (μ.restrict (e ⁻¹' B))) := by
rw [hres.map_eq]
exact hf.restrict
calc
_ = ∫ ω', f ω' ∂Measure.map e (μ.restrict (e ⁻¹' B)) :=
(integral_map hres.measurable.aemeasurable hfm).symm
_ = _ := by rw [hres.map_eq]
theorem condExp_comp_measurePreserving
{Ω Ω' : Type*} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace 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 : Ω' → ℝ} (hf : Integrable f ν) :
μ[f ∘ e | m.comap e] =ᵐ[μ] (ν[f | m]) ∘ e := by
let : MeasurableSpace Ω' := mΩ'
have hsource : m.comap e ≤ mΩ :=
(MeasurableSpace.comap_mono hm).trans he.measurable.comap_le
have hcompint : Integrable ((ν[f | m]) ∘ e) μ :=
he.integrable_comp_of_integrable integrable_condExp
apply Filter.EventuallyEq.symm
apply ae_eq_condExp_of_forall_setIntegral_eq hsource
(he.integrable_comp_of_integrable hf)
(fun _ _ _ => hcompint.integrableOn)
· intro A hA _
obtain ⟨B, hB, rfl⟩ := MeasurableSpace.measurableSet_comap.mp hA
simp only [Function.comp_apply]
rw [setIntegral_comp_measurePreserving μ ν e he (f := ν[f | m])
integrable_condExp.aestronglyMeasurable (hm B hB),
setIntegral_comp_measurePreserving μ ν e he hf.aestronglyMeasurable (hm B hB),
setIntegral_condExp hm hf hB]
· exact (stronglyMeasurable_condExp.comp_measurable
(Measurable.of_comap_le le_rfl)).aestronglyMeasurable
theorem condIndepFun_comp_measurePreserving
{Ω Ω' C D : Type*} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω']
[MeasurableSpace C] [MeasurableSpace D]
[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 : Ω' → C} {g : Ω' → D}
(hf : @Measurable Ω' C mΩ' _ f) (hg : @Measurable Ω' D mΩ' _ g)
(hfg : CondIndepFun m hm f g ν) :
CondIndepFun (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)))
(f ∘ e) (g ∘ e) μ := by
let : MeasurableSpace Ω' := mΩ'
classical
rw [condIndepFun_iff_condExp_inter_preimage_eq_mul
(hf.comp he.measurable) (hg.comp he.measurable)]
intro U V hU hV
have hE := hU.preimage hf
have hH := hV.preimage hg
have hCE := condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator hE)
have hCH := condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator hH)
have hCEH := condExp_comp_measurePreserving μ ν e he m hm
((integrable_const (1 : ℝ)).indicator (hE.inter hH))
have hmul := he.quasiMeasurePreserving.ae_eq_comp
((condIndepFun_iff_condExp_inter_preimage_eq_mul hf hg).mp hfg U V hU hV)
filter_upwards [hCE, hCH, hCEH, hmul] with ω hωE hωH hωEH hωmul
exact hωEH.trans (hωmul.trans (congrArg₂ (· * ·) hωE.symm hωH.symm))
theorem ae_eq_iff_comp_measurePreserving
{Ω Ω' C : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']
[MeasurableSpace C] [MeasurableEq C]
(μ : Measure Ω) (ν : Measure Ω') (e : Ω → Ω')
(he : MeasurePreserving e μ ν) {f g : Ω' → C}
(hf : Measurable f) (hg : Measurable g) :
f =ᵐ[ν] g ↔ f ∘ e =ᵐ[μ] g ∘ e := by
rw [← he.map_eq]
exact ae_map_iff he.measurable.aemeasurable (measurableSet_eq_fun hf hg)
end Problem367
end
end OAI
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