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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | import OAI.Probability.FirstPassage.BKPathMeasurability
namespace OAI
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
end OAI
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