import Mathlib import OAI.Probability.Ballisticity.Estimates.WeightedConditioned namespace OAI section section open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience lemma weak_limit_of_bounded_fresh_map {Ω E : Type*} [MeasurableSpace Ω] [MetricSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] (μ : Measure Ω) [IsProbabilityMeasure μ] (η : ℕ → Measure Ω) [∀ i, IsProbabilityMeasure (η i)] (S : Ω → Ω) (hS : Measurable S) (hmap : ∀ i, (η i).map S = μ) (C : ℝ≥0∞) (hC : C ≠ ⊤) (hdom : ∀ i, η i ≤ C • μ) (F : ℕ → Ω → E) (hF : ∀ i, Measurable (F i)) (W : ProbabilityMeasure E) (hlim : Tendsto (fun i => ProbabilityMeasure.map (⟨μ,inferInstance⟩ : ProbabilityMeasure Ω) (F i)) atTop (𝓝 W)) (hclose : ∀ ε, 0 < ε → Tendsto (fun i => μ.real {ω | ε ≤ dist (F i (S ω)) (F i ω)}) atTop (𝓝 0)) : Tendsto (fun i => ProbabilityMeasure.map (⟨η i,inferInstance⟩ : ProbabilityMeasure Ω) (F i)) atTop (𝓝 W) := by apply weak_map_of_varying_close η (fun i => F i ∘ S) F (fun i => (hF i).comp hS) hF W · have he (i : ℕ) : ProbabilityMeasure.map (⟨η i,inferInstance⟩ : ProbabilityMeasure Ω) (F i ∘ S) = ProbabilityMeasure.map (⟨μ,inferInstance⟩ : ProbabilityMeasure Ω) (F i) := by apply Subtype.ext change (η i).map (F i ∘ S) = μ.map (F i) rw [← Measure.map_map (hF i) hS,hmap] simpa only [he] using hlim · intro ε hε have hb (i : ℕ) : (η i).real {ω | ε ≤ dist (F i (S ω)) (F i ω)} ≤ C.toReal * μ.real {ω | ε ≤ dist (F i (S ω)) (F i ω)} := by have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top hC (measure_ne_top _ _)) (hdom i {ω | ε ≤ dist (F i (S ω)) (F i ω)}) simpa only [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul,Measure.real] using hh exact squeeze_zero (fun _ => measureReal_nonneg) hb (by simpa only [mul_zero] using (hclose ε hε).const_mul C.toReal) lemma weightedConditioned_finite {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) [IsProbabilityMeasure (conditionedLaw ν ℓ)] (g : Environment d → ℝ≥0∞) (hg : Measurable g) (hgle : ∀ ω, g ω ≤ 1) : IsFiniteMeasure (weightedConditioned ν ℓ g) := ⟨lt_of_le_of_lt (weightedConditioned_le ν ℓ g hg hgle Set.univ) (measure_lt_top _ _)⟩ lemma normalized_weightedConditioned_highSuffix {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (J : ℝ) (hJ : 0 ≤ J) (g : Environment d → ℝ≥0∞) (hg : @Measurable _ _ (rowSigma {z | dot (realPosition z) ℓ ≤ J}) _ g) (hgle : ∀ ω, g ω ≤ 1) (hpos : weightedConditioned ν ℓ g Set.univ ≠ 0) : (normalizedMeasure (weightedConditioned ν ℓ g)).map (highSuffix ℓ J) = conditionedLaw ν ℓ := by let : IsProbabilityMeasure (conditionedLaw ν ℓ) := conditionedLaw_probability ν ℓ (ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)) let : IsFiniteMeasure (weightedConditioned ν ℓ g) := weightedConditioned_finite ν ℓ g (hg.mono (rowSigma_le _) le_rfl) hgle rw [normalizedMeasure,Measure.map_smul _ (measurable_highSuffix ℓ J hJ).aemeasurable, weightedConditioned_highSuffix ν ℓ htrans J hJ g hg hgle, smul_smul,ENNReal.inv_mul_cancel hpos (measure_ne_top _ _),one_smul] lemma normalized_weightedConditioned_le {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) (g : Environment d → ℝ≥0∞) (hg : Measurable g) (hgle : ∀ ω, g ω ≤ 1) (c : ℝ≥0∞) (hc : c ≤ weightedConditioned ν ℓ g Set.univ) : normalizedMeasure (weightedConditioned ν ℓ g) ≤ c⁻¹ • conditionedLaw ν ℓ := by intro A change (weightedConditioned ν ℓ g Set.univ)⁻¹ * weightedConditioned ν ℓ g A ≤ c⁻¹ * conditionedLaw ν ℓ A exact mul_le_mul' (ENNReal.inv_le_inv.mpr hc) (weightedConditioned_le ν ℓ g hg hgle A) end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI