Theorems · Theorem · measure theory

MeasureTheory.measure_limsup_cofinite_eq_zero

∀ {α : Type u_1} {ι : Type u_2} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [MeasureTheory.OuterMeasureClass F α]
  [Countable ι] {μ : F} {s : ι → Set α}, ∑' (i : ι), μ (s i) ≠ ⊤ → μ (Filter.limsup s Filter.cofinite) = 0

One direction of the Borel-Cantelli lemma (sometimes called the "first Borel-Cantelli lemma"): if (s i) is a countable family of sets such that ∑' i, μ (s i) is finite, then the limit superior of the s i along the cofinite filter is a null set. Note: for the second Borel-Cantelli lemma (applying to independent sets in a probability space), see ProbabilityTheory.measure_limsup_eq_one.

Defined in

Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Cited by

3 results in Mathlib

Foundations

Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FunLikeMeasureTheory.OuterMeasureClassCountable