| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.GridOscillation |
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| namespace OAI |
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| 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 |
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| 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) |
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| 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))⟩ |
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| end DirectionalTransience |
|
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
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| end |
| end |
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| end OAI |
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