For questions involving the Borel-Cantelli lemma or the second Borel-Cantelli lemma. Use this tag along with (probability-theory), (real-analysis) or (measure-theory).
The Borel-Cantelli lemma in measure theory states that given a sequence $(E_n)$ of measurable sets in a measure space $(S, \Sigma, \mu)$, if the series $$\sum_n \mu(E_n)$$ converges, then $$\mu(\limsup E_n) = 0$$
This means that almost every $x$ in $S$ belongs to $E_n$ for at most finitely many $n$, or, equivalently, that the set of $x$ in $S$ that belong to $E_n$ for infinitely many $n$, has measure $0$.
In probability theory, the first Borel-Cantelli lemma states that, given a sequence $(E_n)$ of events in a probability space $(\Omega, \mathscr F, \mathbb P)$, if the series $$\sum_n \mathbb P(E_n)$$ converges, then $$\mathbb P(\limsup E_n) = 0$$
In probability theory, the second Borel-Cantelli lemma states that, given a sequence $(E_n)$ of independent events in a probability space $(\Omega, \mathscr F, \mathbb P)$, if the series $$\sum_n \mathbb P(E_n)$$ diverges, then $$\mathbb P(\limsup E_n) = 1$$
Given an independent sequence of events $E_1, E_2, ...$ in a probability space $(\Omega, \mathscr F, \mathbb P)$,
$$\mathbb P(\limsup E_n) \in \{ 0,1 \} $$
