Theorems · Theorem · probability

ProbabilityTheory.condExp_zero_or_one_of_measurableSet_limsup

∀ {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
  {β : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : β → Set ι} [inst : StandardBorelSpace Ω] (hm : m ≤ m0)
  [inst_1 : MeasureTheory.IsFiniteMeasure μ],
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.iCondIndep m hm s μ →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x1 x2 => x1 ⊆ x2) ns →
          (∀ (a : β), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) →
              ∀ {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

Defined in

Mathlib.Probability.Independence.ZeroOne

Cited by

0 results in Mathlib

Foundations

Depth 302 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceMeasureTheory.IsFiniteMeasure