Lemma 15.1.7 (First Borel-Cantelli Lemma). Let $(X, \cm, \mu)$ be a measure space and $\seq{E_n}\subset \cm$. If $\sum_{n \in \natp}\mu(E_{n}) < \infty$, then
\[\mu\paren{\limsup_{n \to \infty}E_n}= 0\]
Proof. For any $n \in \natp$, by monotonicity and subadditivity (Proposition 15.1.5),
\[\mu\paren{\limsup_{k \to \infty}E_k}\le \mu\paren{\bigcup_{k \ge n}E_k}\le \sum_{k \ge n}\mu(E_{k})\]
As $\sum_{k \in \natp}\mu(E_{k}) < \infty$,
\[\mu\paren{\limsup_{k \to \infty}E_k}\le \inf_{n \in \natp}\sum_{k \ge n}\mu(E_{k}) = 0\]
$\square$