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For events $B_n\in {\mathcal{F}}_n$, put $p_n=\mathbb{P}(B_n\mid {\mathcal{F}}_{n-1})$. Then ${\sum }_n{1}_{B_n}<\infty$ almost surely on $\{{\sum }_n{p}_n<\infty \}$, while ${\sum }_n{1}_{B_n}=\infty$ almost surely on $\{{\sum }_n{p}_n=\infty \}$. The proof applies the Martingale convergence theorem to compensated sums ${\sum }_{j\le n}(1_{B_j}-p_j)$ and uses Kronecker lemma in the divergent case.

The strong law of large numbers states that for independent identically distributed integrable random variables,To prove it, set $Y_n=X_n{1}_{\{|X_n|\le n\}}$. The tail-sum formula gives ${\sum }_n\mathbb{P}(X_n\ne Y_n)<\infty$, so the Borel-Cantelli lemmas make the two sequences eventually equal. AlsoThe Kolmogorov convergence theorem implies that ${\sum }_n(Y_n-\mathbb{E}{Y}_n)/n$ converges almost surely, and Kronecker lemma yieldsFinally $\mathbb{E}{Y}_n\to \mu$, so the Cesaro mean of these expectations tends to $\mu$.

Set $p_n=\mathbb{P}(B_n\mid {\mathcal{F}}_{n-1})$, $S_n={\sum }_{j\le n}{1}_{B_j}$, and $A_n={\sum }_{j\le n}{p}_j$. Then $M_n=S_n-A_n$ is a martingale with bounded increments and conditional variance at most $p_n$. On $\{A_{\infty }<\infty \}$, localization and the $L^2$ martingale convergence theorem make $M_n$ converge, so the integer-valued increasing sequence $S_n$ is finite. On $\{A_{\infty }=\infty \}$, applying martingale convergence toand Kronecker lemma gives $M_n/A_n\to 0$. Hence $S_n/A_n\to 1$ and $S_n\to \infty$. This is the Conditional Borel-Cantelli lemma, and proves the two events equal almost surely.