| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Set |
| open scoped MeasureTheory |
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| namespace Problem367 |
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| variable {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| [MeasurableSingletonClass A] {μ : Measure Ω} [IsFiniteMeasure μ] |
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| theorem condExp_comap_eq_cond_on_fiber {S : Ω → A} (hS : Measurable S) |
| (a : A) {f : Ω → ℝ} (hf : Integrable f μ) : |
| μ[f | MeasurableSpace.comap S inferInstance] =ᵐ[μ.restrict (S ⁻¹' {a})] |
| fun _ => ∫ ω, f ω ∂μ[|S ⁻¹' {a}] := by |
| let s := S ⁻¹' {a} |
| have hs : MeasurableSet s := hS (measurableSet_singleton a) |
| have hs' : MeasurableSet[MeasurableSpace.comap S inferInstance] s := |
| ⟨{a}, measurableSet_singleton a, rfl⟩ |
| by_cases hμs : μ s = 0 |
| · rw [show S ⁻¹' {a} = s from rfl, Measure.restrict_eq_zero.2 hμs] |
| rfl |
| have havg : (∫ ω, (∫ x, f x ∂μ[|s]) ∂μ.restrict s) = |
| ∫ ω, f ω ∂μ.restrict s := by |
| rw [integral_const, ProbabilityTheory.cond, integral_smul_measure, ENNReal.toReal_inv] |
| simp only [smul_eq_mul, measureReal_def, Measure.restrict_apply_univ] |
| rw [← mul_assoc, mul_inv_cancel₀ (ENNReal.toReal_ne_zero.mpr |
| ⟨hμs, measure_ne_top _ _⟩), one_mul] |
| refine (condExp_restrict_ae_eq_restrict hS.comap_le hs' hf).symm.trans ?_ |
| apply Filter.EventuallyEq.symm |
| apply ae_eq_condExp_of_forall_setIntegral_eq hS.comap_le hf.restrict |
| · intro t _ _ |
| exact (integrable_const _).integrableOn |
| · rintro t ⟨u, hu, rfl⟩ _ |
| rw [Measure.restrict_restrict (hS hu)] |
| by_cases ha : a ∈ u |
| · have heq : S ⁻¹' u ∩ s = s := by |
| ext ω |
| simp only [s, mem_inter_iff, mem_preimage, mem_singleton_iff] |
| exact ⟨And.right, fun h => ⟨h ▸ ha, h⟩⟩ |
| rw [heq] |
| exact havg |
| · have heq : S ⁻¹' u ∩ s = ∅ := by |
| ext ω |
| simp only [s, mem_inter_iff, mem_preimage, mem_singleton_iff, mem_empty_iff_false, |
| iff_false, not_and] |
| intro h hEq |
| exact ha (hEq ▸ h) |
| simp [heq] |
| · exact stronglyMeasurable_const.aestronglyMeasurable |
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| theorem condExp_indicator_comap_eq_cond_on_fiber {S : Ω → A} (hS : Measurable S) |
| (a : A) {t : Set Ω} (ht : MeasurableSet t) : |
| μ⟦t | MeasurableSpace.comap S inferInstance⟧ =ᵐ[μ.restrict (S ⁻¹' {a})] |
| fun _ => μ[|S ⁻¹' {a}].real t := by |
| rw [← integral_indicator_one ht] |
| exact condExp_comap_eq_cond_on_fiber hS a ((integrable_const (1 : ℝ)).indicator ht) |
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| theorem indepFun_cond_fiber_of_condIndepFun [StandardBorelSpace Ω] |
| {B C : Type*} [MeasurableSpace B] [MeasurableSpace C] |
| {S : Ω → A} {X : Ω → B} {Y : Ω → C} |
| (hS : Measurable S) (hX : Measurable X) (hY : Measurable Y) |
| (hXY : CondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X Y μ) |
| (a : A) : IndepFun X Y μ[|S ⁻¹' {a}] := by |
| by_cases hμa : μ (S ⁻¹' {a}) = 0 |
| · rw [cond_eq_zero_of_meas_eq_zero hμa] |
| simp [indepFun_iff_measure_inter_preimage_eq_mul] |
| let : IsProbabilityMeasure μ[|S ⁻¹' {a}] := cond_isProbabilityMeasure hμa |
| have : (ae (μ.restrict (S ⁻¹' {a}))).NeBot := |
| ae_neBot.mpr (mt Measure.restrict_eq_zero.mp hμa) |
| apply indepFun_iff_measure_inter_preimage_eq_mul.mpr |
| intro s t hs ht |
| have hcond := (condIndepFun_iff_condExp_inter_preimage_eq_mul hX hY).mp hXY s t hs ht |
| have hcond' := ae_restrict_le (s := S ⁻¹' {a}) hcond |
| have hleft := condExp_indicator_comap_eq_cond_on_fiber (μ := μ) hS a ((hX hs).inter (hY ht)) |
| have hfirst := condExp_indicator_comap_eq_cond_on_fiber (μ := μ) hS a (hX hs) |
| have hsecond := condExp_indicator_comap_eq_cond_on_fiber (μ := μ) hS a (hY ht) |
| have heq : ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), |
| μ[|S ⁻¹' {a}].real (X ⁻¹' s ∩ Y ⁻¹' t) = |
| μ[|S ⁻¹' {a}].real (X ⁻¹' s) * μ[|S ⁻¹' {a}].real (Y ⁻¹' t) := by |
| filter_upwards [hcond', hleft, hfirst, hsecond] with ω h h₁ h₂ h₃ |
| simpa only [h₁, h₂, h₃] using h |
| obtain ⟨ω, hω⟩ := heq.exists |
| apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (by finiteness)).mp |
| simpa only [measureReal_def, ENNReal.toReal_mul] using hω |
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| theorem iIndepFun_cond_fiber_of_iCondIndepFun [StandardBorelSpace Ω] |
| {ι : Type*} {B : ι → Type*} [mB : ∀ i, MeasurableSpace (B i)] |
| {S : Ω → A} {X : ∀ i, Ω → B i} |
| (hS : Measurable S) (hX : ∀ i, Measurable (X i)) |
| (hXI : iCondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X μ) |
| (a : A) (hμa : μ (S ⁻¹' {a}) ≠ 0) : iIndepFun X μ[|S ⁻¹' {a}] := by |
| classical |
| let : IsProbabilityMeasure μ[|S ⁻¹' {a}] := cond_isProbabilityMeasure hμa |
| have : (ae (μ.restrict (S ⁻¹' {a}))).NeBot := |
| ae_neBot.mpr (mt Measure.restrict_eq_zero.mp hμa) |
| apply iIndepFun_iff_measure_inter_preimage_eq_mul.mpr |
| intro I sets hsets |
| have hcond := (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mB X hX).mp hXI I hsets |
| have hcond' := ae_restrict_le (s := S ⁻¹' {a}) hcond |
| have hmeas : MeasurableSet (⋂ i ∈ I, X i ⁻¹' sets i) := |
| Finset.measurableSet_biInter I (fun i hi => hX i (hsets i hi)) |
| have hleft := condExp_indicator_comap_eq_cond_on_fiber (μ := μ) hS a hmeas |
| have heach : ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), ∀ i ∈ I, |
| (μ⟦X i ⁻¹' sets i | MeasurableSpace.comap S inferInstance⟧) ω = |
| μ[|S ⁻¹' {a}].real (X i ⁻¹' sets i) := by |
| apply (Finset.eventually_all I).mpr |
| intro i hi |
| exact condExp_indicator_comap_eq_cond_on_fiber hS a (hX i (hsets i hi)) |
| have heq : ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), |
| μ[|S ⁻¹' {a}].real (⋂ i ∈ I, X i ⁻¹' sets i) = |
| ∏ i ∈ I, μ[|S ⁻¹' {a}].real (X i ⁻¹' sets i) := by |
| filter_upwards [hcond', hleft, heach] with ω h h₁ h₂ |
| rw [h₁] at h |
| simp only [Finset.prod_apply] at h |
| exact h.trans (Finset.prod_congr rfl h₂) |
| obtain ⟨ω, hω⟩ := heq.exists |
| apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) |
| (ENNReal.prod_ne_top (fun i hi => measure_ne_top _ _))).mp |
| simpa only [measureReal_def, ENNReal.toReal_prod] using hω |
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| theorem condExp_comap_discrete [Countable A] {S : Ω → A} (hS : Measurable S) |
| {f : Ω → ℝ} (hf : Integrable f μ) : |
| μ[f | MeasurableSpace.comap S inferInstance] =ᵐ[μ] |
| fun ω => ∫ x, f x ∂μ[|S ⁻¹' {S ω}] := by |
| have hAll : ∀ᵐ ω ∂μ, ∀ a : A, ω ∈ S ⁻¹' {a} → |
| (μ[f | MeasurableSpace.comap S inferInstance]) ω = |
| ∫ x, f x ∂μ[|S ⁻¹' {a}] := by |
| apply ae_all_iff.mpr |
| intro a |
| exact (ae_restrict_iff' (hS (measurableSet_singleton a))).mp |
| (condExp_comap_eq_cond_on_fiber hS a hf) |
| filter_upwards [hAll] with ω hω |
| exact hω (S ω) rfl |
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| theorem condIndepFun_iff_indepFun_cond_fibers [StandardBorelSpace Ω] [Countable A] |
| {B C : Type*} [MeasurableSpace B] [MeasurableSpace C] |
| {S : Ω → A} {X : Ω → B} {Y : Ω → C} |
| (hS : Measurable S) (hX : Measurable X) (hY : Measurable Y) : |
| CondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X Y μ ↔ |
| ∀ a : A, μ (S ⁻¹' {a}) ≠ 0 → IndepFun X Y μ[|S ⁻¹' {a}] := by |
| constructor |
| · intro h a _ |
| exact indepFun_cond_fiber_of_condIndepFun hS hX hY h a |
| · intro h |
| apply (condIndepFun_iff_condExp_inter_preimage_eq_mul hX hY).mpr |
| intro s t hs ht |
| have hOn (a : A) : ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), |
| (μ⟦X ⁻¹' s ∩ Y ⁻¹' t | MeasurableSpace.comap S inferInstance⟧) ω = |
| (μ⟦X ⁻¹' s | MeasurableSpace.comap S inferInstance⟧) ω * |
| (μ⟦Y ⁻¹' t | MeasurableSpace.comap S inferInstance⟧) ω := by |
| by_cases hμa : μ (S ⁻¹' {a}) = 0 |
| · rw [Measure.restrict_eq_zero.mpr hμa] |
| simp |
| have hprod := (h a hμa).measure_inter_preimage_eq_mul s t hs ht |
| have hreal := congrArg ENNReal.toReal hprod |
| simp only [ENNReal.toReal_mul] at hreal |
| filter_upwards |
| [condExp_indicator_comap_eq_cond_on_fiber hS a ((hX hs).inter (hY ht)), |
| condExp_indicator_comap_eq_cond_on_fiber hS a (hX hs), |
| condExp_indicator_comap_eq_cond_on_fiber hS a (hY ht)] with ω h₁ h₂ h₃ |
| rw [h₁, h₂, h₃] |
| exact hreal |
| have hAll : ∀ᵐ ω ∂μ, ∀ a : A, ω ∈ S ⁻¹' {a} → |
| (μ⟦X ⁻¹' s ∩ Y ⁻¹' t | MeasurableSpace.comap S inferInstance⟧) ω = |
| (μ⟦X ⁻¹' s | MeasurableSpace.comap S inferInstance⟧) ω * |
| (μ⟦Y ⁻¹' t | MeasurableSpace.comap S inferInstance⟧) ω := by |
| apply ae_all_iff.mpr |
| intro a |
| exact (ae_restrict_iff' (hS (measurableSet_singleton a))).mp (hOn a) |
| filter_upwards [hAll] with ω hω |
| exact hω (S ω) rfl |
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| omit [IsFiniteMeasure μ] in |
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| theorem ae_of_ae_restrict_fibers [Countable A] {S : Ω → A} (hS : Measurable S) |
| {p : Ω → Prop} (h : ∀ a : A, ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), p ω) : |
| ∀ᵐ ω ∂μ, p ω := by |
| have hAll : ∀ᵐ ω ∂μ, ∀ a : A, ω ∈ S ⁻¹' {a} → p ω := by |
| apply ae_all_iff.mpr |
| intro a |
| exact (ae_restrict_iff' (hS (measurableSet_singleton a))).mp (h a) |
| filter_upwards [hAll] with ω hω |
| exact hω (S ω) rfl |
|
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| theorem iCondIndepFun_iff_iIndepFun_cond_fibers [StandardBorelSpace Ω] [Countable A] |
| {ι : Type*} {B : ι → Type*} [mB : ∀ i, MeasurableSpace (B i)] |
| {S : Ω → A} {X : ∀ i, Ω → B i} |
| (hS : Measurable S) (hX : ∀ i, Measurable (X i)) : |
| iCondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X μ ↔ |
| ∀ a : A, μ (S ⁻¹' {a}) ≠ 0 → iIndepFun X μ[|S ⁻¹' {a}] := by |
| classical |
| constructor |
| · intro h a hμa |
| exact iIndepFun_cond_fiber_of_iCondIndepFun hS hX h a hμa |
| · intro h |
| apply (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mB X hX).mpr |
| intro I sets hsets |
| apply ae_of_ae_restrict_fibers hS |
| intro a |
| by_cases hμa : μ (S ⁻¹' {a}) = 0 |
| · rw [Measure.restrict_eq_zero.mpr hμa] |
| simp |
| have hprod := (h a hμa).measure_inter_preimage_eq_mul I hsets |
| have hreal := congrArg ENNReal.toReal hprod |
| simp only [ENNReal.toReal_prod] at hreal |
| have hmeas : MeasurableSet (⋂ i ∈ I, X i ⁻¹' sets i) := |
| Finset.measurableSet_biInter I (fun i hi => hX i (hsets i hi)) |
| have heach : ∀ᵐ ω ∂μ.restrict (S ⁻¹' {a}), ∀ i ∈ I, |
| (μ⟦X i ⁻¹' sets i | MeasurableSpace.comap S inferInstance⟧) ω = |
| μ[|S ⁻¹' {a}].real (X i ⁻¹' sets i) := by |
| apply (Finset.eventually_all I).mpr |
| intro i hi |
| exact condExp_indicator_comap_eq_cond_on_fiber hS a (hX i (hsets i hi)) |
| filter_upwards [condExp_indicator_comap_eq_cond_on_fiber hS a hmeas, heach] |
| with ω h₁ h₂ |
| simp only [Finset.prod_apply] |
| rw [h₁, Finset.prod_congr rfl h₂] |
| exact hreal |
|
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| end Problem367 |
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| end |
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| end OAI |
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