On the First and the Second Borel-Cantelli Lemmas

Abstract

Let $\{A_n{\}}_{n=1}^{\infty }$ be a sequence of events and let $S:=\sum _{n=1}^{\infty }{1}_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P}(S<\infty )=1$ and $\mathbb{P}(S=\infty )=1$ respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas.