| import OAI.Probability.FirstPassage.BKPathMeasurability |
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| namespace OAI |
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|
| namespace PlanarFPP |
| open MeasureTheory Filter Set |
| open scoped ENNReal |
|
|
| theorem strict_disjoint_measure_le_of_rounded {Ω : Type*} [MeasurableSpace Ω] |
| (P : Measure Ω) (τ : Edge → Ω → ℝ) (S : Finset Edge) |
| (Fs : List (Set StrictPathWitness)) (C : ℝ≥0∞) |
| (hbound : ∀ t, P {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)} ≤ C) : |
| P {ω | StrictDisjointPaths S Fs (fun e => τ e ω)} ≤ C := by |
| let E : ℕ → Set Ω := fun N => ⋂ t ≥ N, {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)} |
| have he : {ω | StrictDisjointPaths S Fs (fun e => τ e ω)} = ⋃ N, E N := by |
| ext ω |
| simp only [mem_ofPred_eq,mem_iUnion,mem_iInter,E] |
| rw [strict_disjoint_eventual_rounded_iff,eventually_atTop] |
| have hmono : Monotone E := by |
| intro N M hNM ω hω |
| simp only [E,mem_iInter] at hω ⊢ |
| exact fun t ht => hω t (hNM.trans ht) |
| have hE : ∀ N, P (E N) ≤ C := by |
| intro N |
| apply (measure_mono ?_).trans (hbound N) |
| intro ω hω |
| exact mem_iInter.mp (mem_iInter.mp hω N) le_rfl |
| rw [he,hmono.measure_iUnion] |
| exact iSup_le hE |
|
|
| end PlanarFPP |
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| end OAI |
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