| 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 |
|
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| end |
|
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| end OAI |
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|
