Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 24J (Probability and Measure)

A family $(X_n)$ of integrable random variables is uniformly integrable whenEquivalently the integrals of their tails vanish uniformly. This includes uniform $L^1$ boundedness, since $\mathbb{E}|X_n|\le M+\mathbb{E}[|X_n|1_{|X_n|>M}]$.

For conjugate exponents $1<p,q<\infty$, $1/p+1/q=1$, Hölder's inequality is $\boxed{\mathbb{E}|XY|\le \Vert X{\Vert }_p\Vert Y{\Vert }_q}$. If either finite norm is zero, the corresponding variable is zero almost surely. Otherwise normalize $u=|X|/\Vert X{\Vert }_p$, $v=|Y|/\Vert Y{\Vert }_q$. Young inequality $uv\le u^p/p+v^q/q$ follows by minimizing the right side minus $uv$ in $u\ge 0$, whose derivative is $u^{p-1}-v$. Integration gives $\mathbb{E}(uv)\le 1/p+1/q=1$, proving the result. Infinite norms give the usual extended-value inequality whenever the right side is unambiguous. At the endpoints, $\mathbb{E}|XY|\le \Vert X{\Vert }_1\Vert Y{\Vert }_{\infty }$ follows from the almost-sure bound for $Y$, with the symmetric version for $p=\infty$.

$L^p$ 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.

The assertion is false. On $([0,1],\mathcal{B},\text{Lebesgue measure})$, put $X_n=n{1}_{(0,1/n)}$. Then $\mathbb{E}|X_n|=1$ for every $n$, so the family is $L^1$ bounded. For every $M$, choose $n>M$; then $\mathbb{E}[|X_n|1_{|X_n|>M}]=1$. The tail criterion for uniform integrability fails.