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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 | import Mathlib
namespace OAI
section
section
noncomputable section
open MeasureTheory Set ProbabilityTheory
open scoped ENNReal
namespace HardSphere.Comparison
variable {E P W : Type*} [MeasurableSpace E] [MeasurableSpace P] [MeasurableSpace W]
def firstPaths (d : ℕ) (v : Fin (d+2) → E×(P×P)) : Fin (d+1) → P := fun i ↦ (v i.castSucc).2.1
def secondPaths (d : ℕ) (v : Fin (d+2) → E×(P×P)) : Fin (d+1) → P := fun i ↦ (v i.succ).2.2
def observeFirst (d : ℕ) (π : P→E) (p : (Fin (d+2) → E×(P×P))×W) : E×E×(Fin d→E)×W :=
(π (firstPaths d p.1 0),(p.1 (Fin.last (d+1))).1,fun i ↦ π (firstPaths d p.1 i.succ),p.2)
def observeSecond (d : ℕ) (π : P→E) (p : (Fin (d+2) → E×(P×P))×W) : E×E×(Fin d→E)×W :=
((p.1 0).1,π (secondPaths d p.1 (Fin.last d)),fun i ↦ π (secondPaths d p.1 i.castSucc),p.2)
lemma observeFirst_measurable (d : ℕ) {π : P→E} (hπ : Measurable π) : Measurable (observeFirst (W:=W) d π) := by
unfold observeFirst firstPaths
fun_prop
lemma observeSecond_measurable (d : ℕ) {π : P→E} (hπ : Measurable π) : Measurable (observeSecond (W:=W) d π) := by
unfold observeSecond secondPaths
fun_prop
omit [MeasurableSpace E] [MeasurableSpace P] [MeasurableSpace W] in
lemma observations_eq (d : ℕ) (π : P→E) (v : Fin (d+2) → E×(P×P)) (w : W)
(h : ∀k,π (v k).2.1=(v k).1 ∧ π (v k).2.2=(v k).1) :
observeFirst d π (v,w)=observeSecond d π (v,w) := by
unfold observeFirst observeSecond firstPaths secondPaths
simp only [(h _).1,(h _).2]
congr 2
lemma observed_measures_eq (d : ℕ) {π : P→E} (μ : Measure ((Fin (d+2) → E×(P×P))×W))
(h : ∀ᵐ p ∂μ,∀k,π (p.1 k).2.1=(p.1 k).1 ∧ π (p.1 k).2.2=(p.1 k).1) :
μ.map (observeFirst d π)=μ.map (observeSecond d π) := by
apply Measure.map_congr
filter_upwards [h] with p hp
exact observations_eq d π p.1 p.2 hp
lemma conditioned_observed_measures_eq (d : ℕ) {π : P→E}
(μ : Measure ((Fin (d+2) → E×(P×P))×W)) (C : Set ((Fin (d+2) → E×(P×P))×W))
(h : ∀ᵐ p ∂μ,∀k,π (p.1 k).2.1=(p.1 k).1 ∧ π (p.1 k).2.2=(p.1 k).1) :
((μ C)⁻¹ • μ.restrict C).map (observeFirst d π)=
((μ C)⁻¹ • μ.restrict C).map (observeSecond d π) := by
apply observed_measures_eq
exact Measure.smul_absolutelyContinuous.ae_le
(ae_restrict_of_ae h)
end HardSphere.Comparison
end
end
end
end OAI
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