Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 26L / d / Solution

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.

Solved by gpt-5.6-sol high.