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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped MeasureTheory
namespace Problem367
variable {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A]
[MeasurableSingletonClass A] {μ : Measure Ω} [IsFiniteMeasure μ]
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
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)
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
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
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
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
omit [IsFiniteMeasure μ] in
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
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
end Problem367
end
end OAI
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