I came across the following problem in my study of measurable functions:
If $\{f_n\}$ is a sequence of measurable functions on $X$, then $\{x : \lim_{n \to \infty}f_n(x) \mbox{ exists}\}$ is measurable.
I know how to prove this result if we assume the $f_n$ are $\overline{\mathbb{R}}$-valued, since there are particularly nice results about the limit inferior and the limit superior of such functions in this case. I suspect these (lim sup and lim inf) will factor into the more general case, but I was wondering how the proof would proceed!