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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped MeasureTheory
namespace Problem367.KernelExtensionConditionalExpectation
variable {Ω S X : Type*} [mΩ : MeasurableSpace Ω]
[MeasurableSpace S] [mX : MeasurableSpace X]
{μ : Measure Ω} [IsFiniteMeasure μ] {m : MeasurableSpace Ω}
theorem condExp_fst_compProd
(hm : m ≤ mΩ) {p : Ω → S} (hp : Measurable[m] p)
(κ : Kernel S X) [IsMarkovKernel κ] {f : Ω → ℝ} (hf : Integrable f μ) :
let ρ := μ ⊗ₘ κ.comap p (hp.mono hm le_rfl)
ρ[(fun z : Ω × X => f z.1) | m.prod mX] =ᵐ[ρ]
fun z => μ[f | m] z.1 := by
let : MeasurableSpace Ω := mΩ
let : MeasurableSpace (Ω × X) := mΩ.prod mX
let k := κ.comap p (hp.mono hm le_rfl)
let ρ := μ ⊗ₘ k
have hfst : @MeasurePreserving (Ω × X) Ω (mΩ.prod mX) mΩ Prod.fst ρ μ :=
⟨measurable_fst, Measure.fst_compProd μ k⟩
have hF : Integrable (fun z : Ω × X => f z.1) ρ :=
hfst.integrable_comp_of_integrable hf
have hG : Integrable (fun z : Ω × X => μ[f | m] z.1) ρ :=
hfst.integrable_comp_of_integrable integrable_condExp
have hsmall : m.prod mX ≤ mΩ.prod mX :=
sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
have hsm : StronglyMeasurable[m.prod mX]
(fun z : Ω × X => μ[f | m] z.1) :=
stronglyMeasurable_condExp.comp_measurable measurable_fst
have hrect (a : Set Ω) (ha : MeasurableSet[m] a)
(b : Set X) (hb : MeasurableSet b) :
(∫ z in a ×ˢ b, μ[f | m] z.1 ∂ρ) = ∫ z in a ×ˢ b, f z.1 ∂ρ := by
let w : Ω → ℝ := fun ω => (κ (p ω)).real b
have hw : StronglyMeasurable[m] w :=
((κ.measurable_coe hb).ennreal_toReal.comp hp).stronglyMeasurable
have hwb : ∀ᵐ ω ∂μ, ‖w ω‖ ≤ (1 : ℝ) := by
apply Eventually.of_forall
intro ω
rw [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]
exact measureReal_le_one
have hwf : Integrable (w * f) μ :=
hf.bdd_mul (hw.mono hm).aestronglyMeasurable hwb
have he : μ[w * f | m] =ᵐ[μ] w * μ[f | m] :=
condExp_stronglyMeasurable_mul_of_bound hm hw hf 1 hwb
rw [Measure.setIntegral_compProd (hm a ha) hb hG.integrableOn,
Measure.setIntegral_compProd (hm a ha) hb hF.integrableOn]
simp only [setIntegral_const, smul_eq_mul]
change (∫ ω in a, w ω * μ[f | m] ω ∂μ) = ∫ ω in a, w ω * f ω ∂μ
calc
_ = ∫ ω in a, μ[w * f | m] ω ∂μ :=
setIntegral_congr_ae (hm a ha) (he.symm.mono (fun ω hω _ => hω))
_ = _ := setIntegral_condExp hm hwf ha
have hwhole : (∫ z, μ[f | m] z.1 ∂ρ) = ∫ z : Ω × X, f z.1 ∂ρ := by
simpa only [univ_prod_univ, Measure.restrict_univ] using
hrect univ MeasurableSet.univ univ MeasurableSet.univ
refine (ae_eq_condExp_of_forall_setIntegral_eq hsmall hF
(fun a _ _ => hG.integrableOn) ?_ hsm.aestronglyMeasurable).symm
intro a ha _
refine MeasurableSpace.induction_on_inter
(@generateFrom_prod Ω X m mX).symm (@isPiSystem_prod Ω X m mX)
(by simp) ?_ ?_ ?_ a ha
· rintro u ⟨s, hs, t, ht, rfl⟩
exact hrect s hs t ht
· intro a ha he
rw [setIntegral_compl (hsmall a ha) hG,
setIntegral_compl (hsmall a ha) hF, hwhole, he]
· intro a hdisj ha he
rw [integral_iUnion (fun i => hsmall _ (ha i)) hdisj hG.integrableOn,
integral_iUnion (fun i => hsmall _ (ha i)) hdisj hF.integrableOn]
exact tsum_congr he
end Problem367.KernelExtensionConditionalExpectation
end
end OAI
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