After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these exercises.
Perhaps this is an application of Levi?
The question is:
Suppose $X_1,X_2,...$ is a sequence of random variables, not necessarily nonnegative, and $X_n\uparrow X a.s.$. Also assume $\sup_n E(|X_n|)<\infty$.
Show that $E(|X|)<\infty$ and $E(X_n)\rightarrow E(X)$