On the First and the Second Borel-Cantelli Lemmas

Let $\{A_n{\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of events and let ${\displaystyle S:=\sum _{n=1}^{\mathrm{\infty }}{1}_{A_n}}$. We present in this note equivalent characterizations for the statements $\mathbb{P}(S<\mathrm{\infty })=1$ and $\mathbb{P}(S=\mathrm{\infty })=1$ 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.