Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 6 / c

Almost-sure convergence plus uniform integrability gives $X_n\to X$ in $L^1$. Conditional expectation is an $L^1$ contraction, so

Take $X_n=Y_n{Z}_n$ and $\mathcal{G}=\sigma (Y_1,Y_2,\dots )$. Then $\mathbb{P}(X_n=n)=n^{-2}$, so $(X_n)$ is uniformly integrable and $X_n\to 0$ almost surely by the Borel-Cantelli lemmas. Independence and $\mathbb{E}{Z}_n=1$ giveBut the independent events $\{Y_n=1\}$ have divergent probability sum, so the second Borel-Cantelli lemma makes them occur infinitely often. Thus these conditional expectations do not converge almost surely to zero.