Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 26G / a / Solution

Uniform integrability meansIf ${\sup }_n\mathbb{E}|X_n{|}^p=C<\infty$ for $p>1$, thenproving uniform integrability. For a counterexample, let $X_n=n$ with probability $1/n$ and zero otherwise. Then $\mathbb{E}|X_n|=1$, but for every $K$ and $n>K$ the tail expectation is one.