Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 4 / d / Solution

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.