By definition a sequence $\{X_n\}_{n \ge 0}$ is uniformly integrable if $\sup_n E[X_n \cdot \mathbb{I}_{\{X_n > a\}}] \to 0$ and $a \to \infty$.
An equivalent definition is that 1) $\sup_n E[|X_n|] < \infty$ and 2) $\forall \epsilon > 0, \exists \, \delta > 0$ such that for all $n$, $E[X_n \cdot \mathbb{I}_A] < \epsilon$ for any event $A$ such that $P(A) < \delta$.
I'm just curious- can someone give an example of a sequence where $\sup_n E[|X_n|] < \infty$ but $\{X_n\}$ is not uniformly integrable? Every sequence I've tried to create seems to be uniformly integrable.
Thanks.
Best Answer
Assume your probability space is $([0,1],dx)$, with $dx$ denoting the Lesbegue measure.
Let $X_n=n 1_{[0,1/n]}$. Then $\mathbb{E}|X_n|=1$ for every $n$ but $(X_n)_ {n\in \mathbb{N}}$ is not uniformly integrable as $$ \mathbb{E}|X_n| \mathbb{I}_{|X_n|>a}=\mathbb{E}|X_n| $$ for all $n>a$.