Theorems · Theorem · probability

ProbabilityTheory.condExp_zero_or_one_of_measurableSet_limsup_atBot

∀ {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
  [inst : SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] [inst_3 : StandardBorelSpace Ω] (hm : m ≤ m0)
  [inst_4 : MeasureTheory.IsFiniteMeasure μ],
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.iCondIndep m hm s μ →
      ∀ {t : Set Ω},
        MeasurableSet t → ∀ᵐ (ω : Ω) ∂μ, μ[t.indicator fun ω => 1 | m] ω = 0 ∨ μ[t.indicator fun ω => 1 | m] ω = 1

Kolmogorov's 0-1 law, conditional version: any event in the tail σ-algebra of a conditionally independent sequence of sub-σ-algebras has conditional probability 0 or 1.

Defined in

Mathlib.Probability.Independence.ZeroOne

Cited by

0 results in Mathlib

Foundations

Depth 303 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemilatticeInfNoMinOrderNonemptyStandardBorelSpaceMeasureTheory.IsFiniteMeasure