= Conditional Borel-Cantelli lemma
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Let $(\mathcal F_n)$ be a <filtration (probability theory)>[filtration]. For <event>[events] $B_n\in\mathcal F_n$, put $p_n=\mathbb P(B_n\mid\mathcal F_{n-1})$. Then $\sum_n\mathbf1_{B_n}<\infty$ <almost surely> on $\{\sum_np_n<\infty\}$, while $\sum_n\mathbf1_{B_n}=\infty$ <almost surely> on $\{\sum_np_n=\infty\}$. The proof applies the <martingale convergence theorem> to compensated sums $\sum_{j\leq n}(\mathbf1_{B_j}-p_j)$ and uses <Kronecker lemma> in the divergent case.
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