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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 65 66 | import Mathlib
import OAI.Analysis.HardSphere.Comparison.FiberNormalize
import OAI.Analysis.HardSphere.Conditioning.RouteObservation
namespace OAI
section
section
noncomputable section
open MeasureTheory Set Filter ProbabilityTheory
open scoped ENNReal
namespace HardSphere.Comparison
lemma fiberNormalize_supported {W V : Type*} [MeasurableSpace W] [MeasurableSpace V]
(κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)} (hC : MeasurableSet C) (w : W) :
∀ᵐ v ∂fiberNormalize κ C w,(w,v)∈C := by
change ∀ᵐ v ∂(fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w,(w,v)∈C
apply Measure.smul_absolutelyContinuous.ae_le
rw [fiberRestrict_apply κ hC]
exact ae_restrict_mem (hC.preimage (measurable_const.prodMk measurable_id))
def commonLaw {E P W : Type*} [MeasurableSpace E] [MeasurableSpace P] [MeasurableSpace W]
(d : ℕ) (ξ : Measure W) [SFinite ξ]
(τ : Measure (Fin (d+2) → E×(P×P))) [IsFiniteMeasure τ]
(C : Set (W×(Fin (d+2) → E×(P×P)))) : Measure (W×(Fin (d+2) → E×(P×P))) :=
ξ.compProd (fiberNormalize (Kernel.const W τ) C)
lemma commonLaw_environment {E P W : Type*} [MeasurableSpace E] [MeasurableSpace P] [MeasurableSpace W]
(d : ℕ) (ξ : Measure W) [SFinite ξ]
(τ : Measure (Fin (d+2) → E×(P×P))) [IsFiniteMeasure τ]
{C : Set (W×(Fin (d+2) → E×(P×P)))} (hC : MeasurableSet C)
(hpos : ∀ᵐ w ∂ξ,0<τ {v | (w,v)∈C}) : (commonLaw d ξ τ C).map Prod.fst=ξ :=
fiberNormalize_fst ξ (Kernel.const W τ) hC hpos
lemma commonLaw_probability {E P W : Type*} [MeasurableSpace E] [MeasurableSpace P] [MeasurableSpace W]
(d : ℕ) (ξ : Measure W) [IsProbabilityMeasure ξ]
(τ : Measure (Fin (d+2) → E×(P×P))) [IsFiniteMeasure τ]
{C : Set (W×(Fin (d+2) → E×(P×P)))} (hC : MeasurableSet C)
(hpos : ∀ᵐ w ∂ξ,0<τ {v | (w,v)∈C}) : IsProbabilityMeasure (commonLaw d ξ τ C) :=
⟨fiberNormalize_compProd_mass ξ (Kernel.const W τ) hC hpos⟩
lemma commonLaw_sameObservation {E P W : Type*} [MeasurableSpace E] [MeasurableEq E]
[MeasurableSpace P] [MeasurableSpace W] (d : ℕ) (π : P→E) (hπ : Measurable π)
(ξ : Measure W) [SFinite ξ]
(τ : Measure (Fin (d+2) → E×(P×P))) [IsFiniteMeasure τ]
(hτ : ∀ᵐ v ∂τ,∀k,π (v k).2.1=(v k).1 ∧ π (v k).2.2=(v k).1)
{C : Set (W×(Fin (d+2) → E×(P×P)))} (hC : MeasurableSet C) :
(commonLaw d ξ τ C).map (fun p ↦ observeFirst d π p.swap)=
(commonLaw d ξ τ C).map (fun p ↦ observeSecond d π p.swap) := by
apply Measure.map_congr
have h : ∀ᵐ p ∂commonLaw d ξ τ C,∀k,π (p.2 k).2.1=(p.2 k).1 ∧ π (p.2 k).2.2=(p.2 k).1 := by
rw [commonLaw,Measure.ae_compProd_iff (by measurability)]
exact ae_of_all _ (fun w ↦ (fiberNormalize_ac (Kernel.const W τ) hC w).ae_le hτ)
filter_upwards [h] with p hp
exact observations_eq d π p.2 p.1 hp
end HardSphere.Comparison
end
end
end
end OAI
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