Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 26L (Probability and Measure)

Fatou says $\int \mathrm{liminf}{f}_n\le \mathrm{liminf}\int f_n$ for nonnegative functions. Apply it to $g+f_n$ and $g-f_n$ when $|f_n|\le g\in L^1$ to obtain both inequalities and hence dominated convergence.

For every $ϵ>0$, $1_{|X_n-X|>ϵ}\to 0$ almost surely and is dominated by 1, so its expectation, the corresponding probability, tends to zero.

Yes. The $L^2$ bound makes the family uniformly integrable in $L^1$: $E[|X_n|1_{|X_n|>K}]\le E[X_n^2]/K\le 1/K$. Together with convergence in probability, Vitali's theorem gives $E|X_n|\to 0$.

On disjoint events $A_n$ with $P(A_n)=c/n^2$, set $X_n=n{1}_{A_n}$. Then $X_n\to 0$ almost surely and $E{X}_n^2=c$, but any common dominator has integral at least ${\sum }_{n}nP(A_n)=c\sum 1/n=\infty$. Thus no integrable dominating variable exists.