Locally of ae
ProbabilityTheory.locally_of_ae
Plain-language statement
If the filtration satisfies the usual conditions, then a property of the paths of a process that holds almost surely holds locally.
Exact Lean statement
lemma locally_of_ae [𝓕.IsComplete P] {p : (ι → E) → Prop} (hpX : ∀ᵐ ω ∂P, p (X · ω))
(hp₀ : p (0 : ι → E)) :
Locally (fun X ↦ ∀ ω, p (X · ω)) 𝓕 X PFormal artifact
Lean source
lemma locally_of_ae [𝓕.IsComplete P] {p : (ι → E) → Prop} (hpX : ∀ᵐ ω ∂P, p (X · ω)) (hp₀ : p (0 : ι → E)) : Locally (fun X ↦ ∀ ω, p (X · ω)) 𝓕 X P := by refine ⟨_, isLocalizingSequence_localizingSequenceOfProp hpX, fun _ ω ↦ ?_⟩ by_cases hω : p (X · ω) · convert hω using 2 rw [stoppedProcess_eq_of_le, Set.indicator_of_mem] · simp [LocalizingSequenceOfProp, if_pos hω] · simp [LocalizingSequenceOfProp, if_pos hω] · convert hp₀ using 2 rw [stoppedProcess_eq_of_ge, Set.indicator_of_notMem] · rfl · simp [LocalizingSequenceOfProp, if_neg hω] · simp [LocalizingSequenceOfProp, if_neg hω]- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/Locally.lean:55-68
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.