CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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First Borel-Cantelli lemma for events

Statement

Let $(A_{n})_{n \in N}$ be events in a probability space. If $\sum_{n = 0}^{\infty} P (A_{n}) < + \infty,$ then $P (A_{n} i . o .) = 0.$

No independence hypothesis is needed.

Facts & Assumptions

Given: Events $(A_{n})_{n \in N}$ with $\sum_{n = 0}^{\infty} P (A_{n}) < + \infty$ .

[L1]

The infinitely-often event is the set limsup. (Limsup and the infinitely often event)

[L2]

If the sum of the measures is finite, then the measure of the set limsup is zero. (The first Borel-Cantelli lemma for measures)

Proof

technique · direct

1.1

By [L1], the event ${A_{n} i . o .}$ is exactly ${lim sup}_{n \to \infty} A_{n}$ .

2.1

Applying [L2] to the probability measure $P$ and the measurable sets $A_{n}$ gives $P (A_{n} i . o .) = P ({lim sup}_{n \to \infty} A_{n}) = 0.$

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.3.1 (standard reference, not scraped)