| import OAI.NumberTheory.DukePrimeDegree.Geometry.CountableWeakTests |
| import Mathlib.Topology.Sequences |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| A countable family of finite-basis-union evaluations defines a stronger |
| second-countable topology on probabilities. Kernel evaluations are measurable |
| for that topology, and the countable Portmanteau criterion makes the map |
| to the usual weak topology continuous. No global measurable-space |
| instance on probabilities is changed. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set TopologicalSpace |
| open scoped Topology ENNReal |
|
|
| namespace DukePrimeDegree |
|
|
| variable (Y : Type*) [TopologicalSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace Y] [OpensMeasurableSpace Y] |
|
|
| /-- A separate wrapper retains both probability topologies without |
| overriding the existing weak topology. -/ |
| structure BasisProbability where |
| toProbability : ProbabilityMeasure Y |
|
|
| def basisProbabilityTests (p : BasisProbability Y) : |
| Finset (countableBasis Y) → ℝ≥0∞ := |
| fun F => (p.toProbability : Measure Y) (finiteBasisUnion F) |
| |
| instance : TopologicalSpace (BasisProbability Y) := |
| TopologicalSpace.induced (basisProbabilityTests Y) inferInstance |
|
|
| omit [OpensMeasurableSpace Y] in |
| lemma basisProbabilityTests_isInducing : |
| Topology.IsInducing (basisProbabilityTests Y) := ⟨rfl⟩ |
| |
| instance : SecondCountableTopology (BasisProbability Y) := |
| (basisProbabilityTests_isInducing Y).secondCountableTopology |
| |
| instance : MeasurableSpace (BasisProbability Y) := borel (BasisProbability Y) |
| |
| instance : BorelSpace (BasisProbability Y) := ⟨rfl⟩ |
|
|
| theorem continuous_basisProbability_toProbability : |
| Continuous (BasisProbability.toProbability (Y := Y)) := by |
| apply continuous_iff_seqContinuous.mpr |
| intro u p hu |
| apply tendsto_probability_of_finiteBasisUnion |
| intro F |
| exact ((continuous_apply F).comp (basisProbabilityTests_isInducing Y).continuous).continuousAt.tendsto.comp hu |
|
|
| variable {Y} |
|
|
| theorem measurable_kernel_basisProbability |
| {Z : Type*} [MeasurableSpace Z] (κ : Kernel Z Y) [IsMarkovKernel κ] : |
| Measurable (fun z => |
| (⟨⟨κ z, inferInstance⟩⟩ : BasisProbability Y)) := by |
| have hm : Measurable (fun z => basisProbabilityTests Y |
| (⟨⟨κ z, inferInstance⟩⟩ : BasisProbability Y)) := by |
| apply measurable_pi_iff.mpr |
| intro F |
| exact measurable_kernel_finiteBasisUnion κ F |
| apply Measurable.of_comap_le |
| change MeasurableSpace.comap (fun z => |
| (⟨⟨κ z, inferInstance⟩⟩ : BasisProbability Y)) |
| (@borel (BasisProbability Y) (TopologicalSpace.induced (basisProbabilityTests Y) _)) ≤ _ |
| rw [borel_comap, MeasurableSpace.comap_comp, |
| ← BorelSpace.measurable_eq (α := Finset (countableBasis Y) → ℝ≥0∞)] |
| exact hm.comap_le |
|
|
| /-- The codomain carries the Borel sigma algebra of weak convergence. -/ |
| theorem measurable_kernel_probability_weak |
| {Z : Type*} [mZ : MeasurableSpace Z] (κ : Kernel Z Y) [IsMarkovKernel κ] : |
| @Measurable Z (ProbabilityMeasure Y) mZ (borel (ProbabilityMeasure Y)) |
| (fun z => ⟨κ z, inferInstance⟩) := by |
| have hcont := (continuous_basisProbability_toProbability Y).borel_measurable |
| exact hcont.comp (measurable_kernel_basisProbability κ) |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
