TITLE: Finite almost surely implies integrable?
QUESTION [8 upvotes]: 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 $\mathbb{E}[X]<\infty$?

REPLY [11 votes]: No.
Take $P(X=2^n)=2^{-n}$.
Then $P(X<\infty)=1$ and $E(X)=\infty$.