File size: 2,465 Bytes
67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Bridges.LocalReference
import OAI.Analysis.HardSphere.Conditioning.ProductNormalize
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.Comparison
variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
lemma conditioned_map_preimage (μ : Measure X) (f : X→Y) (hf : Measurable f)
{S : Set Y} (hS : MeasurableSet S) :
(conditionedMeasure μ (f⁻¹' S)).map f=conditionedMeasure (μ.map f) S := by
unfold conditionedMeasure
rw [Measure.map_smul _ hf.aemeasurable,Measure.map_apply hf hS]
congr 1
ext B hB
rw [Measure.map_apply hf hB,Measure.restrict_apply (hB.preimage hf),
Measure.restrict_apply hB,Measure.map_apply hf (hB.inter hS)]
rfl
lemma conditioned_map_ae (μ : Measure X) (f : X→Y) (hf : Measurable f)
(E : Set X) {S : Set Y} (hS : MeasurableSet S) (hE : E=ᵐ[μ] f⁻¹' S) :
(conditionedMeasure μ E).map f=conditionedMeasure (μ.map f) S := by
have he : conditionedMeasure μ E=conditionedMeasure μ (f⁻¹' S) := by
unfold conditionedMeasure
rw [measure_congr hE,Measure.restrict_congr_set hE]
rw [he]
exact conditioned_map_preimage μ f hf hS
lemma conditioned_pi_map {I : Type*} [Fintype I] {X Y : I→Type*}
[∀i,MeasurableSpace (X i)] [∀i,MeasurableSpace (Y i)]
(μ : ∀i,Measure (X i)) [∀i,IsFiniteMeasure (μ i)]
(f : ∀i,X i→Y i) (hf : ∀i,Measurable (f i))
{S : Set (∀i,Y i)} (hS : MeasurableSet S) :
(conditionedMeasure (Measure.pi μ) ((fun x i ↦ f i (x i))⁻¹' S)).map
(fun x i ↦ f i (x i))=
conditionedMeasure (Measure.pi fun i ↦ (μ i).map (f i)) S := by
have hmf : Measurable (fun x : ∀i,X i ↦ fun i ↦ f i (x i)) := by
exact Measurable.of_eval (fun i ↦ (hf i).comp (measurable_pi_apply i))
rw [conditioned_map_preimage _ _ hmf hS,Measure.pi_map_pi (fun i ↦ (hf i).aemeasurable)]
end HardSphere.Comparison
namespace HardSphere.Bridge
open Wiener GaussianPacking Comparison
lemma noise_reference_identification (t : ℝ) (y z : Point) (E : Set Noise)
{S : Set C(Time,Point)} (hS : MeasurableSet S)
(hE : E=ᵐ[law] (leg (2*t) y z)⁻¹' S) :
(conditionedMeasure law E).map (leg (2*t) y z)=individualReference t y z S :=
conditioned_map_ae law _ (leg_continuous _ _ _).measurable E hS hE
end HardSphere.Bridge
end
end
end
end OAI
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