Is a sequence of bounded random variables uniformly integrable?

Question

It seems to me that a sequence of uniformly bounded random variables $\{X_n\}_{n=1}^\infty$ on a probability space $(\Omega_n, \mathcal{F}_n, P)$ such that $|X_n(\omega)| \leq M$ for all $\omega \in \Omega_n$ is uniform integrable. For if $I$ is the indicator function, suppose $\varepsilon > 0$ is given and let $k = 2M$, then $$ E(|X_n| I(|X_n| \geq k)) = E(|X_n| I(|X_n| \geq 2M)) = E(|X_n|\cdot 0) = 0 < \varepsilon $$ Is it true that a sequence of uniformly bounded random variables are uniformly integrable?

Answer 1

The reasoning and its conclusion are correct.