| import Mathlib.MeasureTheory.Measure.Portmanteau |
| import Mathlib.Probability.Kernel.Basic |
| import Mathlib.Topology.Bases |
|
|
| namespace OAI |
|
|
| /-! |
| # Countably many measure evaluations determine weak convergence |
|
|
| Finite unions from a countable open basis suffice for the Portmanteau |
| lower bound on every open set. This is used to connect the measurable |
| kernel evaluations to the weak topology, without assuming a measurable |
| selection of tangent frames. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory Filter Set TopologicalSpace |
| open scoped Topology |
|
|
| namespace DukePrimeDegree |
|
|
| variable {Y : Type*} [TopologicalSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace Y] [OpensMeasurableSpace Y] |
|
|
| def finiteBasisUnion (F : Finset (countableBasis Y)) : Set Y := |
| ⋃ U ∈ F, (U.val : Set Y) |
|
|
| omit [MeasurableSpace Y] [OpensMeasurableSpace Y] in |
| lemma isOpen_finiteBasisUnion (F : Finset (countableBasis Y)) : |
| IsOpen (finiteBasisUnion F) := |
| isOpen_biUnion fun U _ => isOpen_of_mem_countableBasis U.property |
|
|
| theorem tendsto_probability_of_finiteBasisUnion |
| (μs : ℕ → ProbabilityMeasure Y) (μ : ProbabilityMeasure Y) |
| (h : ∀ F : Finset (countableBasis Y), |
| Tendsto (fun k => (μs k : Measure Y) (finiteBasisUnion F)) atTop |
| (𝓝 ((μ : Measure Y) (finiteBasisUnion F)))) : |
| Tendsto μs atTop (𝓝 μ) := by |
| classical |
| apply tendsto_of_forall_isOpen_le_liminf_nat' |
| intro U hU |
| let I := {B : countableBasis Y // (B.val : Set Y) ⊆ U} |
| let W : Finset I → Set Y := fun F => ⋃ B ∈ F, (B.val.val : Set Y) |
| have hWU (F : Finset I) : W F ⊆ U := by |
| intro x hx |
| simp only [W, mem_iUnion] at hx |
| obtain ⟨B, hB, hxB⟩ := hx |
| exact B.property hxB |
| have hmono : Monotone W := by |
| intro F G hFG x hx |
| simp only [W, mem_iUnion] at hx ⊢ |
| obtain ⟨B, hB, hxB⟩ := hx |
| exact ⟨B, hFG hB, hxB⟩ |
| have hunion : (⋃ F : Finset I, W F) = U := by |
| apply subset_antisymm |
| · exact iUnion_subset hWU |
| · intro x hx |
| obtain ⟨B, hB, hxB, hBU⟩ := |
| (isBasis_countableBasis Y).exists_subset_of_mem_open hx hU |
| let i : I := ⟨⟨B, hB⟩, hBU⟩ |
| refine mem_iUnion.mpr ⟨{i}, ?_⟩ |
| exact mem_iUnion.mpr ⟨i, mem_iUnion.mpr ⟨by simp, hxB⟩⟩ |
| have hWbasis (F : Finset I) : W F = finiteBasisUnion (F.image fun B => B.val) := by |
| ext x |
| simp only [W, finiteBasisUnion, mem_iUnion, Finset.mem_image] |
| constructor |
| · rintro ⟨B, hB, hx⟩ |
| exact ⟨B.val, ⟨B, hB, rfl⟩, hx⟩ |
| · rintro ⟨B, ⟨C, hC, rfl⟩, hx⟩ |
| exact ⟨C, hC, hx⟩ |
| have hmeasure : (μ : Measure Y) U = ⨆ F : Finset I, (μ : Measure Y) (W F) := by |
| rw [← hunion, hmono.measure_iUnion] |
| rw [hmeasure] |
| apply iSup_le |
| intro F |
| have hconv : Tendsto (fun k => (μs k : Measure Y) (W F)) atTop |
| (𝓝 ((μ : Measure Y) (W F))) := by |
| simpa only [hWbasis] using h (F.image fun B => B.val) |
| calc |
| (μ : Measure Y) (W F) = liminf (fun k => (μs k : Measure Y) (W F)) atTop := |
| hconv.liminf_eq.symm |
| _ ≤ liminf (fun k => (μs k : Measure Y) U) atTop := |
| liminf_le_liminf (Eventually.of_forall fun k => measure_mono (hWU F)) |
|
|
| /-- The coordinate map into the countable family of open-set tests is |
| measurable for every actual probability kernel. -/ |
| theorem measurable_kernel_finiteBasisUnion |
| {Z : Type*} [MeasurableSpace Z] (κ : ProbabilityTheory.Kernel Z Y) |
| (F : Finset (countableBasis Y)) : |
| Measurable (fun z => κ z (finiteBasisUnion F)) := |
| ProbabilityTheory.Kernel.measurable_coe κ (isOpen_finiteBasisUnion F).measurableSet |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
