First Borel–Cantelli lemma

Statement

Let $A_1, A_2, \dots$ be events in a probability space $(\Omega, \mathcal{F}, P)$ with $\sum_{n=1}^{\infty} P(A_n) < \infty$. Then $$P\left(\limsup_{n \to \infty} A_n\right) = 0, \quad \text{where} \quad \limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_m.$$ So almost surely, only finitely many of the events $A_n$ occur.

Remarks

The event $\limsup_n A_n$ is the set of outcomes that lie in infinitely many $A_n$. It is written $\{A_n \text{ i.o.}\}$, for "infinitely often". Proof: for each $n$, $P(\limsup_m A_m) \le P\left(\bigcup_{m \ge n} A_m\right) \le \sum_{m \ge n} P(A_m)$ by Boole's inequality. The right side tends to $0$ as $n \to \infty$. The lemma needs no independence. The second Borel–Cantelli lemma gives a partial converse for independent events. The lemma holds in each measure space: if $\sum_n \mu(A_n) < \infty$, then $\mu(\limsup_n A_n) = 0$.

History

The lemma is named after Émile Borel and Francesco Paolo Cantelli.

Explanations

L1 human

If the chances of the events add up to a finite number, then after some time none of them happens any more, with probability $1$. For example, if $P(A_n) = 1/n^2$, then only finitely many $A_n$ occur.