| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset Function Preorder |
| open scoped ENNReal |
| namespace Problem367.CountableProjective |
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| variable {X : ℕ → Type*} [∀ n, MeasurableSpace (X n)] |
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| 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⟩) |
|
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| private lemma split_measurable (n : ℕ) : Measurable (split (X := X) n) := by |
| exact (measurable_frestrictLe₂ _).prodMk (measurable_pi_apply _) |
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|
| 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 |
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| 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 |
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| variable [∀ n, StandardBorelSpace (X n)] [∀ n, Nonempty (X n)] |
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| 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 |
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| 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⟩ |
|
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| end Problem367.CountableProjective |
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| end |
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| end OAI |
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|
