import Mathlib import OAI.Probability.Ballisticity.Geometry.HeightPolygonTranslateApply import OAI.Probability.Ballisticity.Walk.CompactPathModulus 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 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 heightPolygon_random_shift_small {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (F : ℕ → ℕ → Ω → ℝ) (hF : ∀ i k, Measurable (F i k)) (r n : ℕ → ℝ) (hn : Tendsto n atTop atTop) (H : Ω → ℕ) (hH : Measurable H) (c : ℕ → Ω → ℝ) (hc : TendstoInMeasure μ (fun i ω => c i ω/r i) atTop 0) {T : ℝ} (hT : 0 ≤ T) (ht : IsTightMeasureSet (Set.range (fun i => μ.map (fun ω => heightPolygon (fun k => F i k ω) (r i) (n i) (T+1))))) : TendstoInMeasure μ (fun i ω => heightPolygon (fun k => F i (k+H ω) ω+c i ω) (r i) (n i) T- heightPolygon (fun k => F i k ω) (r i) (n i) T) atTop 0 := by have hhn : TendstoInMeasure μ (fun i ω => (H ω:ℝ)/n i) atTop 0 := by apply tendstoInMeasure_of_tendsto_ae (fun i => (((measurable_of_countable (fun k : ℕ => (k : ℝ))).comp hH).div_const _).aestronglyMeasurable) exact ae_of_all _ (fun _ => tendsto_const_nhds.div_atTop hn) apply tendstoInMeasure_iff_measureReal_norm.2 intro ε hε simp only [Pi.zero_apply,sub_zero] apply Metric.tendsto_nhds.2 intro β hβ obtain ⟨δ,hδ,hmod⟩ := tight_path_modulus _ ht (by positivity : 0 < ε/4) (by positivity : 0 < β/4) have hht := (tendstoInMeasure_iff_measureReal_norm.1 hhn) (min δ 1) (by positivity) have hct := (tendstoInMeasure_iff_measureReal_norm.1 hc) (ε/4) (by positivity) simp only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] at hht hct filter_upwards [hn.eventually (eventually_gt_atTop (0:ℝ)), hht.eventually_lt_const (by positivity : 0 < β/4), hct.eventually_lt_const (by positivity : 0 < β/4)] with i hni hhi hci let A := {ω | min δ 1 ≤ |(H ω:ℝ)/n i|} let B := {ω | heightPolygon (fun k => F i k ω) (r i) (n i) (T+1) ∈ ContinuousOscillation (ε/4) δ} let C := {ω | ε/4 ≤ |c i ω/r i|} have hb : μ.real B ≤ β/4 := by have hm := hmod i rw [Measure.map_apply (measurable_heightPolygon _ (hF i) _ _ _) (isOpen_continuousOscillation _ _).measurableSet] at hm exact ENNReal.toReal_le_of_le_ofReal (by positivity) hm have hs : {ω | ε ≤ ‖heightPolygon (fun k => F i (k+H ω) ω+c i ω) (r i) (n i) T- heightPolygon (fun k => F i k ω) (r i) (n i) T‖} ⊆ A ∪ B ∪ C := by intro ω hω by_contra hh have ha : ω ∉ A := fun h => hh (Or.inl (Or.inl h)) have hb' : ω ∉ B := fun h => hh (Or.inl (Or.inr h)) have hc' : ω ∉ C := fun h => hh (Or.inr h) have ha' : (H ω:ℝ)/n i < min δ 1 := by simpa only [A,Set.mem_ofPred_eq,not_le,abs_of_nonneg (div_nonneg (Nat.cast_nonneg _) hni.le)] using ha have hHi : (H ω:ℝ) ≤ n i := ((div_lt_one hni).mp (ha'.trans_le (min_le_right _ _))).le have hnorm := heightPolygon_translate_norm_le (fun k => F i k ω) (r i) (n i) T hni hT (H ω) hHi (c i ω) (ε/4) δ (ha'.le.trans (min_le_left _ _)) hb' have hcc : |c i ω/r i| < ε/4 := lt_of_not_ge hc' simp only [Set.mem_ofPred_eq] at hω linarith have hm := (measureReal_mono (μ := μ) hs (measure_ne_top _ _)).trans ((measureReal_union_le (μ := μ) (s₁ := A ∪ B) (s₂ := C)).trans (add_le_add (measureReal_union_le (μ := μ) (s₁ := A) (s₂ := B)) le_rfl)) rw [Real.dist_eq,sub_zero,abs_of_nonneg measureReal_nonneg] dsimp only [A,C] at hm linarith end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI