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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace Problem367
theorem exists_eq_dirac_of_selfProd_ae_eq
{X : Type*} [MeasurableSpace X] (nu : Measure X)
[IsProbabilityMeasure nu]
(h : ∀ᵐ p ∂nu.prod nu, p.1 = p.2) :
∃ x : X, nu = Measure.dirac x := by
obtain ⟨x, hx⟩ := (Measure.ae_ae_of_ae_prod h).exists
refine ⟨x, ?_⟩
have hconst : (fun y : X => y) =ᵐ[nu] (fun _ => x) := by
filter_upwards [hx] with y hy using hy.symm
simpa using Measure.map_congr hconst
theorem exists_measurable_dirac_of_forall
{W X : Type*} [MeasurableSpace W] [MeasurableSpace X]
(kappa : Kernel W X)
(h : ∀ w, ∃ x : X, kappa w = Measure.dirac x) :
∃ F : W → X, Measurable F ∧ ∀ w, kappa w = Measure.dirac (F w) := by
choose F hF using h
refine ⟨F, ?_, hF⟩
intro s hs
have heq : F ⁻¹' s = (fun w => kappa w s) ⁻¹' {1} := by
ext w
simp only [mem_preimage, mem_singleton_iff, hF w, Measure.dirac_apply' _ hs]
exact (indicator_eq_one_iff_mem ENNReal).symm
rw [heq]
exact kappa.measurable_coe hs (measurableSet_singleton 1)
theorem exists_measurable_dirac_of_selfProd_ae_eq
{W X : Type*} [MeasurableSpace W] [MeasurableSpace X]
[MeasurableEq X] [Nonempty X]
(mu : Measure W) (kappa : Kernel W X) [IsMarkovKernel kappa]
(h : ∀ᵐ w ∂mu, ∀ᵐ p ∂(kappa w).prod (kappa w), p.1 = p.2) :
∃ F : W → X, Measurable F ∧ ∀ᵐ w ∂mu, kappa w = Measure.dirac (F w) := by
classical
let good : Set W := {w | (kappa ×ₖ kappa) w (diagonal X)ᶜ = 0}
have hgood : MeasurableSet good :=
((kappa ×ₖ kappa).measurable_coe measurableSet_diagonal.compl)
(measurableSet_singleton 0)
have hgood_ae : ∀ᵐ w ∂mu, w ∈ good := by
filter_upwards [h] with w hw
change (kappa ×ₖ kappa) w (diagonal X)ᶜ = 0
rw [Kernel.prod_apply]
exact hw
let base : X := Classical.choice inferInstance
let modified := Kernel.piecewise hgood kappa
(Kernel.deterministic (fun _ : W => base) measurable_const)
have hmodified : ∀ w, ∃ x : X, modified w = Measure.dirac x := by
intro w
by_cases hw : w ∈ good
· have hdiag : ∀ᵐ p ∂(kappa w).prod (kappa w), p.1 = p.2 := by
change (kappa w).prod (kappa w) {p | ¬p.1 = p.2} = 0
simpa only [good, mem_ofPred_eq, Kernel.prod_apply, diagonal, compl_ofPred] using hw
obtain ⟨x, hx⟩ := exists_eq_dirac_of_selfProd_ae_eq (kappa w) hdiag
exact ⟨x, by simpa [modified, Kernel.piecewise_apply, Kernel.deterministic_apply, hw] using hx⟩
· exact ⟨base, by simp [modified, Kernel.piecewise_apply, Kernel.deterministic_apply, hw]⟩
obtain ⟨F, hF, hdirac⟩ := exists_measurable_dirac_of_forall modified hmodified
refine ⟨F, hF, ?_⟩
filter_upwards [hgood_ae] with w hw
simpa [modified, Kernel.piecewise_apply, Kernel.deterministic_apply, hw] using hdirac w
theorem ae_eq_recovery_of_ae_dirac
{W X : Type*} [MeasurableSpace W] [MeasurableSpace X]
[MeasurableEq X] (mu : Measure W) (kappa : Kernel W X)
{F : W → X} (hF : Measurable F)
(h : ∀ᵐ w ∂mu, kappa w = Measure.dirac (F w)) :
∀ᵐ p ∂mu ⊗ₘ kappa, p.2 = F p.1 := by
apply Measure.ae_compProd_of_ae_ae (measurableSet_eq_fun measurable_snd (hF.comp measurable_fst))
filter_upwards [h] with w hw
rw [hw]
simp
theorem exists_measurable_recovery_of_conditionalCopies
{W X : Type*} [MeasurableSpace W] [MeasurableSpace X]
[MeasurableEq X] [Nonempty X]
(mu : Measure W) [SFinite mu] (kappa : Kernel W X) [IsMarkovKernel kappa]
(h : ∀ᵐ p ∂mu ⊗ₘ (kappa ×ₖ kappa), p.2.1 = p.2.2) :
∃ F : W → X, Measurable F ∧
(∀ᵐ w ∂mu, kappa w = Measure.dirac (F w)) ∧
∀ᵐ p ∂mu ⊗ₘ kappa, p.2 = F p.1 := by
have hcopies : ∀ᵐ w ∂mu, ∀ᵐ p ∂(kappa w).prod (kappa w), p.1 = p.2 := by
simpa only [Kernel.prod_apply] using Measure.ae_ae_of_ae_compProd h
obtain ⟨F, hF, hdirac⟩ := exists_measurable_dirac_of_selfProd_ae_eq mu kappa hcopies
exact ⟨F, hF, hdirac, ae_eq_recovery_of_ae_dirac mu kappa hF hdirac⟩
end Problem367
end
end OAI
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