This question is about the equivalent definition of uniformly integrable, which
A possible uncountable collection of random variables $\{X_{\alpha}, \alpha\in I\}$ is said to be uniformly integrable if $$\lim_{M\longrightarrow\infty}\sup_{\alpha}\mathbb{E}\Big(|X_{\alpha}|\mathbb{1}_{|X_{\alpha}|>M}\Big)=0$$
However, I have seen many times from either books or solutions that they conclude the uniform integrability of a sequence of random variables $\{X_{n}\}$ from the result that $\sup_{n}|X_{n}|$ is integrable.
Is this a equivalent definition or is it just sufficient condition? Why?
Thank you!
Best Answer
The condition is sufficient but not necessary. On $(0,1)$ with Lebesgue measure let $X_n=n^{3/2} I_{(\frac 1 {n+1}, \frac 1 n)}$. Then $sup_n X_n \geq n^{3/2}$ on $(\frac 1 {n+1}, \frac 1 n)$ for every $n$ which makes $E( sup X_n)= \infty$. However $EX_n \to 0$ which implies uniform integrability.