Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 24J / c / Solution

$L^p$ boundedness means ${\sup }_n\Vert X_n{\Vert }_p<\infty$, with the essential-supremum norm for $p=\infty$. For finite $p>1$, let $C={\sup }_n\mathbb{E}|X_n{|}^p<\infty$. On $|X_n|>M$, $|X_n|\le M^{1-p}|X_n{|}^p$, soThis proves uniform integrability. For an $L^{\infty }$ bounded family the tails are identically zero beyond the common essential bound.