Almost-sure convergence plus <uniform integrability> gives $X_n\to X$ in $L^1$. Conditional expectation is an $L^1$ contraction, so
$$
\mathbb E|\mathbb E[X_n\mid\mathcal G]-\mathbb E[X\mid\mathcal G]|
\leq\mathbb E|X_n-X|\to0.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/6/c/i/solution