Theorems · Theorem · probability

ProbabilityTheory.IsStable.locally_induction

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

If q is a stable property and p implies q locally, then p locally implies q locally.

Defined in

Mathlib.Probability.Process.LocalProperty

Cited by

1 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderBotTopologicalSpaceOrderTopologySecondCountableTopologyMeasureTheory.IsFiniteMeasureDenselyOrderedNoMaxOrderZeroMeasureTheory.Filtration.IsRightContinuous