Theorems · Theorem · probability

ProbabilityTheory.IsStable.locally_locally_iff

∀ {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} [SecondCountableTopology ι]
  [MeasureTheory.IsFiniteMeasure P] [DenselyOrdered ι] [NoMaxOrder ι] [inst_7 : Zero E] [𝓕.IsRightContinuous],
  ProbabilityTheory.IsStable 𝓕 p →
    (ProbabilityTheory.Locally (fun Y => ProbabilityTheory.Locally p 𝓕 Y P) 𝓕 X P ↔ ProbabilityTheory.Locally p 𝓕 X P)

A stable property holding locally is idempotent.

Defined in

Mathlib.Probability.Process.LocalProperty

Cited by

1 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderBotTopologicalSpaceOrderTopologySecondCountableTopologyMeasureTheory.IsFiniteMeasureDenselyOrderedNoMaxOrderZeroMeasureTheory.Filtration.IsRightContinuous