Existing in Mathlib · Probability

Second Borel–Cantelli Lemma

If independent measurable events carry an infinite total amount of probability, then with probability one an outcome belongs to infinitely many of them.

measurable events
mutual independence
divergent measure sum
set limsup
infinitely often

Exact theorem

Exact Mathlib statement

theorem ProbabilityTheory.measure_limsup_eq_one {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s Filter.atTop) = 1

The theorem at a glance

Second Borel–Cantelli Lemma at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory probability poster shows an endless sequence of independent event windows, a divergent accumulated-measure column, and almost every trajectory returning through infinitely many windows. — Independent measurable events with divergent total measure occur infinitely often with measure one.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

An unending sequence of independent event apertures recedes across a forest-green field while almost every gold trajectory crosses infinitely many arbitrarily late apertures. — Divergent independent event mass forces almost-sure recurrence through arbitrarily late events.

Why it matters

A mathematical landmark

Borel–Cantelli turns a divergent series of event probabilities plus independence into an almost-sure recurrence law. It is a central bridge between measure estimates and long-run probabilistic behavior.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's second Borel–Cantelli lemma for a mutually independent sequence of measurable sets whose measures have divergent sum. It concludes that the set-theoretic limsup has measure one. It does not assert the first Borel–Cantelli direction, replace mutual independence by pairwise independence, give a convergence rate, or identify a finite stage at which every point belongs.

The hypothesis is the recorded mutual-independence predicate iIndepSet, not merely an informal pairwise-independence condition.
Measure one for the limsup does not mean that every point lies in every sufficiently late event.
This selected declaration is the second lemma; the summable first Borel–Cantelli direction is separate.

Source and local evidence

Where the theorem comes from

Existing declaration: ProbabilityTheory.measure_limsup_eq_one in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.second-borel-cantelli-lemma.v001
Source: Open the pinned upstream reference