| 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 |
| |
