Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and let $(A_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of events. Define

an event by the closure properties of \ref{def:sigma-algebra-measurable-space-2026a}; it consists of exactly those $\omega\in\Omega$ that belong to $A_m$ for infinitely many $m$.

\textbf{First Borel–Cantelli lemma.} If $\sum_{m}P(A_m)<\infty$ (sum as in \ref{def:measure-measure-space-2026a}), then

\textbf{Second Borel–Cantelli lemma.} If the events $(A_m)$ are \reftext{def:independence-events-rvs-2026a}{independent} and $\sum_m P(A_m)=\infty$, then