Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $E(X_n)\rightarrow E(X)$

share|cite|improve this question
Can't you just apply monotone convergence to $Y_n = X_n - X_1$? – Nate Eldredge Jul 28 '12 at 16:16
up vote 2 down vote accepted

First, Fatou's lemma gives $E (\liminf_n |X_n|) \leq \liminf_n E |X_n| \leq \sup_n E |X_n| < \infty$. Since $\liminf_n |X_n| = |X|$ a.e., this shows that $E |X| < \infty$.

We have $0 \leq X - X_n \leq X - X_1 $ a.e. We know that $E |X-X_1| \leq E |X| + E |X_1| < \infty$, so apply the DCT to get $\lim_n E (X - X_n) = 0$, from which the result follows.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.