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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Finset Function Preorder
open scoped ENNReal
namespace Problem367.CountableProjective
variable {X : ℕ → Type*} [∀ n, MeasurableSpace (X n)]
private def split (n : ℕ) (x : ∀ i : Iic (n+1), X i) :
(∀ i : Iic n, X i) × X (n+1) :=
(frestrictLe₂ (Nat.le_succ n) x, x ⟨n+1, mem_Iic.mpr le_rfl⟩)
private lemma split_measurable (n : ℕ) : Measurable (split (X := X) n) := by
exact (measurable_frestrictLe₂ _).prodMk (measurable_pi_apply _)
omit [∀ n, MeasurableSpace (X n)] in
private lemma split_injective (n : ℕ) : Injective (split (X := X) n) := by
intro x y h
funext i
by_cases hi : i.val ≤ n
· exact congrFun (congrArg Prod.fst h) ⟨i.val, mem_Iic.mpr hi⟩
· have hi' : i.val = n+1 := by have := mem_Iic.mp i.property; omega
have he : i = ⟨n+1, mem_Iic.mpr le_rfl⟩ := Subtype.ext hi'
subst i
exact congrArg Prod.snd h
omit [∀ n, MeasurableSpace (X n)] in
private lemma split_comp_restrict (n : ℕ) :
split (X := X) n ∘ frestrictLe (n+1) =
fun x : ∀ n, X n => (frestrictLe n x, x (n+1)) := rfl
variable [∀ n, StandardBorelSpace (X n)] [∀ n, Nonempty (X n)]
theorem exists_nat_marginals
(μ : (n : ℕ) → Measure (∀ i : Iic n, X i))
[∀ n, IsProbabilityMeasure (μ n)]
(hμ : ∀ n, (μ (n+1)).map (frestrictLe₂ (Nat.le_succ n)) = μ n) :
∃ ν : Measure (∀ n, X n), IsProbabilityMeasure ν ∧
∀ n, ν.map (frestrictLe n) = μ n := by
let κ : (n : ℕ) → Kernel (∀ i : Iic n, X i) (X (n+1)) :=
fun n => condDistrib (fun x : ∀ i : Iic (n+1), X i =>
x ⟨n+1, mem_Iic.mpr le_rfl⟩) (frestrictLe₂ (Nat.le_succ n)) (μ (n+1))
let (n : ℕ) : IsMarkovKernel (κ n) := by dsimp [κ]; infer_instance
let ν : Measure (∀ n, X n) := (Kernel.traj κ 0) ∘ₘ μ 0
have hν : IsProbabilityMeasure ν := by dsimp [ν]; infer_instance
refine ⟨ν, hν, ?_⟩
have hstep (n : ℕ) :
(ν.map (frestrictLe n)) ⊗ₘ κ n =
ν.map (fun x => (frestrictLe n x, x (n+1))) := by
rw [Measure.compProd_eq_comp_prod]
dsimp [ν]
rw [Measure.map_comp _ _ (by fun_prop), Kernel.traj_map_frestrictLe,
Measure.comp_assoc, Measure.map_comp _ _ (by fun_prop)]
congr with x₀ : 1
rw [Kernel.comp_apply, ← Measure.compProd_eq_comp_prod,
Kernel.map_apply _ (by fun_prop), Kernel.partialTraj_compProd_eq_map_traj (Nat.zero_le n)]
intro n
induction n with
| zero =>
dsimp [ν]
rw [Measure.map_comp _ _ (measurable_frestrictLe 0),
Kernel.traj_map_frestrictLe, Kernel.partialTraj_self, Measure.id_comp]
| succ n ih =>
apply (split_measurable (X := X) n).measurableEmbedding (split_injective n) |>.map_injective
rw [Measure.map_map (split_measurable n) (measurable_frestrictLe _)]
rw [split_comp_restrict, ← hstep, ih]
have hc := compProd_map_condDistrib (mβ := inferInstance)
(μ := μ (n+1)) (X := frestrictLe₂ (Nat.le_succ n))
(Y := fun x : ∀ i : Iic (n+1), X i => x ⟨n+1, mem_Iic.mpr le_rfl⟩)
(measurable_frestrictLe₂ _).aemeasurable
(measurable_pi_apply _).aemeasurable
rw [hμ n] at hc
exact hc
theorem exists_nat_projective_limit
(μ : (I : Finset ℕ) → Measure (∀ i : I, X i))
[∀ I, IsProbabilityMeasure (μ I)] (hμ : IsProjectiveMeasureFamily μ) :
∃ ν : Measure (∀ n, X n), IsProbabilityMeasure ν ∧ IsProjectiveLimit ν μ := by
obtain ⟨ν, hν, hmarg⟩ := exists_nat_marginals (fun n => μ (Iic n))
(fun n => (hμ (Iic (n+1)) (Iic n) (Iic_subset_Iic.mpr (Nat.le_succ n))).symm)
exact ⟨ν, hν, (isProjectiveLimit_nat_iff hμ ν).mpr hmarg⟩
end Problem367.CountableProjective
end
end OAI
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