File size: 3,293 Bytes
a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Geometry.CountableWeakTests
import Mathlib.Topology.Sequences
namespace OAI
/-!
# A probability kernel is measurable for the weak topology
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
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