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

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.

If a random variable $X$ satisfies $\mathbb{P}[X<\infty]=1$ (which means $X$ is finite almost surely, doesn't it?) then $\mathbb{E}[X]<\infty$?

share|cite|improve this question
up vote 7 down vote accepted

No. Take $P(X=2^n)=2^{-n}$. Then $P(X<\infty)=1$ and $E(X)=\infty$.

share|cite|improve this answer
Could you please expand your answer to give additional conditions under the statement in the question will be true? – seek Feb 14 at 13:41
@seek The best condition I can think of is .... that $\Bbb E[X]<\infty$. – Hagen von Eitzen Feb 14 at 14:08

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.