 Conditional Borel-Cantelli lemma

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.