Theorems · Theorem · probability

ProbabilityTheory.IsStable.locally_of_isPreLocalizingSequence

∀ {ι : Type u_1} {Ω : Type u_2} {E : Type u_3} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [inst : ConditionallyCompleteLinearOrderBot ι] [inst_1 : TopologicalSpace ι] [inst_2 : OrderTopology ι]
  {𝓕 : MeasureTheory.Filtration ι mΩ} {X : ι → Ω → E} {p : (ι → Ω → E) → Prop} [inst_3 : Zero E] [DenselyOrdered ι]
  [FirstCountableTopology ι] [NoMaxOrder ι] {τ : ℕ → Ω → WithTop ι},
  ProbabilityTheory.IsStable 𝓕 p →
    ∀ [𝓕.IsRightContinuous],
      ProbabilityTheory.IsPreLocalizingSequence 𝓕 τ P →
        (∀ (n : ℕ), p (MeasureTheory.stoppedProcess (fun i => {ω | ⊥ < τ n ω}.indicator (X i)) (τ n))) →
          ProbabilityTheory.Locally p 𝓕 X P

A process X satisfies a stable property p locally if there exists a pre-localizing sequence τ for which the stopped processes of fun i ↦ {ω | ⊥ < τ n ω}.indicator (X i) satisfy p.

Defined in

Mathlib.Probability.Process.LocalProperty

Cited by

1 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderBotTopologicalSpaceOrderTopologyZeroDenselyOrderedFirstCountableTopologyNoMaxOrderMeasureTheory.Filtration.IsRightContinuous