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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?

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    Yes, exactly. The finiteness of $Y$ plays a key role in this proof.2012-10-24

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