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

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

As X \geq 0 then X_n \leq X_{n+1} (trivial to show) for all n \in \mathbb{N} where 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 \lim_{n \rightarrow \infty} X_n = X.