I'm faced with the following problem as part of a larger homework problem and do not really know how to go about it:
- let $X_n$ be an increasing sequence of random variables (that could be negative).
- assume that $\lim_{n\rightarrow\infty}X_n = X$ a.e.
- assume there exists a random variable $Y$ such that $X_n \geq Y$ for all $n \in \mathbb{N}$ a.e.
- assume that $\mathbb{E}[X_n] < \infty$ for all $n$, $\mathbb{E}[X] < \infty$, and $\mathbb{E}[Y] < \infty$
With these assumptions in place, I'm trying to show that $\lim_{n\rightarrow\infty}\mathbb{E}[X_n] = \mathbb{E}[X]$, but I can't do it because everything I know about random variables requires that either the $X_n$ or the $Y$ are non-negative.
Anyone with an idea of how to proceed?