Theorems · Theorem · probability

MeasureTheory.StronglyAdapted.stoppedProcess

∀ {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [inst : TopologicalSpace β]
  [TopologicalSpace.PseudoMetrizableSpace β] [inst_2 : Nonempty ι] [inst_3 : LinearOrder ι]
  [inst_4 : TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι] [inst_7 : MeasurableSpace ι]
  [BorelSpace ι] {f : MeasureTheory.Filtration ι m} {u : ι → Ω → β} {τ : Ω → WithTop ι}
  [TopologicalSpace.MetrizableSpace ι],
  MeasureTheory.StronglyAdapted f u →
    (∀ (ω : Ω), Continuous fun i => u i ω) →
      MeasureTheory.IsStoppingTime f τ → MeasureTheory.StronglyAdapted f (MeasureTheory.stoppedProcess u τ)

The stopped process of a strongly adapted process with continuous paths is strongly adapted.

Defined in

Mathlib.Probability.Process.Stopping

Cited by

0 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.PseudoMetrizableSpaceNonemptyLinearOrderTopologicalSpaceSecondCountableTopologyOrderTopologyMeasurableSpaceBorelSpaceTopologicalSpace.MetrizableSpace