| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| [IsProbabilityMeasure μ] {X : Ω → ℝ} | |
| theorem integral_sq_sub_copies_eq_two_variance (hX : MemLp X 2 μ) : | |
| (∫ p : Ω × Ω, (X p.1 - X p.2) ^ 2 ∂μ.prod μ) = | |
| 2 * variance X μ := by | |
| have h₁ : MemLp (fun p : Ω × Ω => X p.1) 2 (μ.prod μ) := hX.comp_fst μ | |
| have h₂ : MemLp (fun p : Ω × Ω => X p.2) 2 (μ.prod μ) := hX.comp_snd μ | |
| have hm : (∫ p : Ω × Ω, (X p.1 - X p.2) ∂μ.prod μ) = 0 := by | |
| rw [integral_sub (h₁.integrable (by simp)) (h₂.integrable (by simp))] | |
| simp [integral_fun_fst, integral_fun_snd] | |
| have hv := variance_of_integral_eq_zero | |
| (X := fun p : Ω × Ω => X p.1 - X p.2) (h₁.sub h₂).aemeasurable hm | |
| rw [← hv] | |
| rw [variance_fun_sub h₁ h₂, covariance_fst_snd_prod hX hX, | |
| measurePreserving_fst.variance_fun_comp hX.aemeasurable, | |
| measurePreserving_snd.variance_fun_comp hX.aemeasurable] | |
| ring | |
| theorem integral_sq_sub_copies (hX : MemLp X 2 μ) : | |
| (∫ p : Ω × Ω, (X p.1 - X p.2) ^ 2 ∂μ.prod μ) = | |
| 2 * ((∫ x, X x ^ 2 ∂μ) - (∫ x, X x ∂μ) ^ 2) := by | |
| rw [integral_sq_sub_copies_eq_two_variance hX, variance_eq_sub hX] | |
| rfl | |
| theorem ae_eq_integral_of_copies_ae_eq (hX : MemLp X 2 μ) | |
| (hcopy : ∀ᵐ p : Ω × Ω ∂μ.prod μ, X p.1 = X p.2) : | |
| ∀ᵐ x ∂μ, X x = ∫ y, X y ∂μ := by | |
| apply ae_eq_integral_of_variance_eq_zero hX | |
| have hz : (∫ p : Ω × Ω, (X p.1 - X p.2) ^ 2 ∂μ.prod μ) = 0 := by | |
| apply integral_eq_zero_of_ae | |
| filter_upwards [hcopy] with p hp | |
| simp [hp] | |
| rw [integral_sq_sub_copies_eq_two_variance hX] at hz | |
| linarith | |
| section Kernel | |
| variable {W : Type*} [MeasurableSpace W] {ν : Measure W} [SFinite ν] | |
| {κ : Kernel W Ω} [IsMarkovKernel κ] | |
| theorem ae_memLp_two_of_compProd {Y : W × Ω → ℝ} | |
| (hY : MemLp Y 2 (ν ⊗ₘ κ)) : | |
| ∀ᵐ w ∂ν, MemLp (fun x => Y (w, x)) 2 (κ w) := by | |
| have hm : ∀ᵐ w ∂ν, AEStronglyMeasurable (fun x => Y (w, x)) (κ w) := | |
| AEStronglyMeasurable.ae_of_compProd (f := fun w x => Y (w, x)) | |
| (by | |
| convert hY.aestronglyMeasurable using 1 | |
| funext p | |
| cases p | |
| rfl) | |
| have hi : ∀ᵐ w ∂ν, Integrable (fun x => Y (w, x) ^ 2) (κ w) := | |
| ((Measure.integrable_compProd_iff hY.integrable_sq.aestronglyMeasurable).1 | |
| hY.integrable_sq).1 | |
| filter_upwards [hm, hi] with w hmw hiw | |
| exact (memLp_two_iff_integrable_sq hmw).2 hiw | |
| theorem exists_measurable_recovery_of_copies_ae_eq (hXm : Measurable X) | |
| (hX : ∀ᵐ w ∂ν, MemLp X 2 (κ w)) | |
| (hcopy : ∀ᵐ z : W × (Ω × Ω) ∂ν ⊗ₘ (κ ×ₖ κ), X z.2.1 = X z.2.2) : | |
| ∃ F : W → ℝ, Measurable F ∧ | |
| ∀ᵐ z : W × Ω ∂ν ⊗ₘ κ, X z.2 = F z.1 := by | |
| let F : W → ℝ := fun w => ∫ x, X x ∂κ w | |
| have hF : Measurable F := hXm.stronglyMeasurable.integral_kernel.measurable | |
| refine ⟨F, hF, ?_⟩ | |
| apply Measure.ae_compProd_of_ae_ae | |
| (measurableSet_eq_fun (hXm.comp measurable_snd) (hF.comp measurable_fst)) | |
| have hc := Measure.ae_ae_of_ae_compProd hcopy | |
| filter_upwards [hX, hc] with w hw hcw | |
| apply ae_eq_integral_of_copies_ae_eq hw | |
| simpa [Kernel.prod_apply] using hcw | |
| theorem exists_measurable_recovery_of_memLp_copies_ae_eq (hXm : Measurable X) | |
| (hX : MemLp (fun z : W × Ω => X z.2) 2 (ν ⊗ₘ κ)) | |
| (hcopy : ∀ᵐ z : W × (Ω × Ω) ∂ν ⊗ₘ (κ ×ₖ κ), X z.2.1 = X z.2.2) : | |
| ∃ F : W → ℝ, Measurable F ∧ | |
| ∀ᵐ z : W × Ω ∂ν ⊗ₘ κ, X z.2 = F z.1 := | |
| exists_measurable_recovery_of_copies_ae_eq hXm (ae_memLp_two_of_compProd hX) hcopy | |
| theorem exists_measurable_recovery_array_of_copies_ae_eq | |
| {ι : Type*} [Countable ι] (Y : ι → Ω → ℝ) | |
| (hYm : ∀ i, Measurable (Y i)) | |
| (hY : ∀ i, ∀ᵐ w ∂ν, MemLp (Y i) 2 (κ w)) | |
| (hcopy : ∀ᵐ z : W × (Ω × Ω) ∂ν ⊗ₘ (κ ×ₖ κ), | |
| ∀ i, Y i z.2.1 = Y i z.2.2) : | |
| ∃ F : W → (ι → ℝ), Measurable F ∧ | |
| ∀ᵐ z : W × Ω ∂ν ⊗ₘ κ, ∀ i, Y i z.2 = F z.1 i := by | |
| have hi : ∀ i, ∃ F : W → ℝ, Measurable F ∧ | |
| ∀ᵐ z : W × Ω ∂ν ⊗ₘ κ, Y i z.2 = F z.1 := by | |
| intro i | |
| apply exists_measurable_recovery_of_copies_ae_eq (hYm i) (hY i) | |
| filter_upwards [hcopy] with z hz using hz i | |
| choose F hFm hF using hi | |
| refine ⟨fun w i => F i w, Measurable.of_eval hFm, ?_⟩ | |
| exact ae_all_iff.2 hF | |
| end Kernel | |
| end Problem367 | |
| end | |
| end OAI | |