I have been lately thinking of how to proof the following:

Let $\displaystyle X: \Omega \rightarrow \Re$ be a random variable and $\displaystyle X \geq 0$ then $\displaystyle \sum_{i=1}^\infty x_i P(X = x)$ converges to the expected value of $\displaystyle X: \Omega \rightarrow \Re$. The only thing that I am given is that $\displaystyle E[X] := \int_\Omega X(\omega)d P(\omega)$ provided that $\displaystyle \int_\Omega |X(\omega)| dP(\omega) < \infty$. Moreover, X is Borel measurable.

As $\displaystyle X \geq 0$ then $\displaystyle X_n \leq X_{n+1}$ (trivial to show) for all $\displaystyle n \in \mathbb{N}$ where $\displaystyle X_n = \sum_{i=1}^n x_i$. After going through all sorts of stuff, I have a feeling that this relevant, but I cannot prove that $\displaystyle \lim_{n \rightarrow \infty} X_n = X$.