Sum of positive random variables

Question

Suppose $(X_n)_{n}$ is a sequence of positive random variables. Suppose $$ \sum_{n \geq 0} E[X_n] < \infty. $$ Can we conclude that $\sum X_n < \infty$ almost surely? If we have independence should be true, I was wondering if it is true also without independence.

Answer 1

Let $X:=\sum_{n=0}^\infty X_n$. Since the $X_n$ are a.s. positive, $X$ is as well, and it follows from Tonelli's theorem that $$ \infty\cdot \mathbb P(X=\infty)\leqslant\mathbb E[X]=\sum_{n=0}^\infty \mathbb E[X_n] <\infty, $$ and hence $\mathbb P(X=\infty)=0$ (under the usual convention that $0\cdot\infty =0$ for the extended real line).