Borel-Cantelli Lemmas

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(A_m)_{m\in\mathbb{N}}$ be a sequence of events. Define

an event by the closure properties of Sigma-Algebra and Measurable Space; it consists of exactly those $\omega\in\Omega$ that belong to $A_m$ for infinitely many $m$.

First Borel–Cantelli lemma. If $\sum_{m}P(A_m)<\infty$ (sum as in Measure, Measure Space, and Probability Measure), then

Second Borel–Cantelli lemma. If the events $(A_m)$ are independent and $\sum_m P(A_m)=\infty$, then