File size: 2,551 Bytes
9b6e014 | 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
import OAI.Probability.Ballisticity.Estimates.SelectedEndpointMoment
import OAI.Probability.Ballisticity.Walk.SelectedFiniteKernel
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology Classical
namespace DirectionalTransience
lemma normalizeFinite_map {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A]
(μ : Measure Ω) (Y : Ω → A) (hY : Measurable Y) :
normalizeFinite (μ.map Y) = (normalizeFinite μ).map Y := by
simp only [normalizeFinite,Measure.map_apply hY MeasurableSet.univ,Set.preimage_univ,
Measure.map_smul _ hY.aemeasurable]
lemma normalizeFinite_restrict {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (N : Set Ω) : normalizeFinite (μ.restrict N) = μ[|N] := by
simp only [normalizeFinite,Measure.restrict_apply_univ,ProbabilityTheory.cond]
lemma selectedFiniteKernel_map_eq {Ω A : Type*} [MeasurableSpace Ω] [MeasurableSpace A]
[MeasurableSingletonClass A] (μ : Measure Ω) (N : Set Ω) (Z : Ω → A) (hZ : Measurable Z)
(E : Set A) (hE : MeasurableSet E) (Y : A → ℝ) (hY : Measurable Y)
(dummy : A) (failure strict : Prop) [Decidable failure] [Decidable strict] :
(selectedFiniteKernel ((μ.restrict N).map Z) E dummy failure strict).map Y =
selectedEndpointLaw μ N (N ∩ Z ⁻¹' E) (Y ∘ Z) (Y dummy) failure strict := by
by_cases hf : failure
· simp only [selectedFiniteKernel,selectedEndpointLaw,ite_eq_left hf,Measure.map_dirac' hY]
simp only [selectedFiniteKernel,selectedEndpointLaw,ite_eq_right hf]
by_cases hs : strict
· simp only [ite_eq_left hs]
rw [Measure.restrict_map hZ hE,Measure.restrict_restrict (hZ hE)]
rw [Set.inter_comm (Z ⁻¹' E) N,normalizeFinite_map _ Z hZ,
normalizeFinite_restrict,Measure.map_map hY hZ]
· simp only [ite_eq_right hs]
rw [normalizeFinite_map _ Z hZ,normalizeFinite_restrict,Measure.map_map hY hZ]
lemma selectedEndpointLaw_cond {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (C N E : Set Ω)
(hC : MeasurableSet C) (hN : MeasurableSet N) (hE : MeasurableSet E)
(Y : Ω → ℝ) (b : ℝ) (failure strict : Prop) [Decidable failure] [Decidable strict] :
selectedEndpointLaw μ[|C] N E Y b failure strict =
selectedEndpointLaw μ (C ∩ N) (C ∩ E) Y b failure strict := by
unfold selectedEndpointLaw
split_ifs with hf hs
· rfl
· rw [cond_cond_eq_cond_inter hC hE]
· rw [cond_cond_eq_cond_inter hC hN]
end DirectionalTransience
end
end
end OAI
|