import Mathlib import OAI.Probability.Ballisticity.Estimates.GridOscillation 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 namespace DirectionalTransience lemma compact_path_modulus (K : Set C(unitInterval,ℝ)) (hK : IsCompact K) {ε : ℝ} (hε : 0 < ε) : ∃ δ, 0 < δ ∧ Disjoint K (ContinuousOscillation ε δ) := by let : CompactSpace K := isCompact_iff_compactSpace.mp hK have hc : Continuous (fun p : K × unitInterval => p.1.1 p.2) := continuous_eval.comp ((continuous_subtype_val.comp continuous_fst).prodMk continuous_snd) obtain ⟨δ,hδ,hm⟩ := Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous hc) ε hε refine ⟨δ/2,by positivity,Set.disjoint_left.mpr ?_⟩ rintro f hf ⟨s,t,hst,hbig⟩ have hd : dist ((⟨f,hf⟩ : K),s) ((⟨f,hf⟩ : K),t) < δ := by simpa only [Prod.dist_eq,dist_self,max_eq_right dist_nonneg] using hst.trans_lt (half_lt_self hδ) have hh := hm hd exact (not_lt_of_ge hbig.le) (by simpa only [Real.dist_eq] using hh) lemma tight_path_modulus (μ : ℕ → Measure C(unitInterval,ℝ)) (ht : IsTightMeasureSet (Set.range μ)) {ε β : ℝ} (hε : 0 < ε) (hβ : 0 < β) : ∃ δ, 0 < δ ∧ ∀ i, μ i (ContinuousOscillation ε δ) ≤ ENNReal.ofReal β := by obtain ⟨K,hK,hKm⟩ := isTightMeasureSet_iff_exists_isCompact_measure_compl_le.mp ht (ENNReal.ofReal β) (ENNReal.ofReal_pos.mpr hβ) obtain ⟨δ,hδ,hd⟩ := compact_path_modulus K hK hε have hsub : ContinuousOscillation ε δ ⊆ Kᶜ := by intro f hf hfK exact Set.disjoint_left.mp hd hfK hf exact ⟨δ,hδ,fun i => (measure_mono hsub).trans (hKm _ (Set.mem_range_self i))⟩ end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI