Theorems · Theorem · probability

MeasureTheory.StronglyAdapted.stoppedProcess_of_discrete

∀ {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} [DiscreteTopology ι],
  MeasureTheory.StronglyAdapted f u →
    MeasureTheory.IsStoppingTime f τ → MeasureTheory.StronglyAdapted f (MeasureTheory.stoppedProcess u τ)

If the indexing order has the discrete topology, then the stopped process of a strongly adapted process is strongly adapted.

Defined in

Mathlib.Probability.Process.Stopping

Cited by

1 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.PseudoMetrizableSpaceNonemptyLinearOrderTopologicalSpaceSecondCountableTopologyOrderTopologyMeasurableSpaceBorelSpaceDiscreteTopology