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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | 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
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