TITLE: Limit a.e. of a sequence measurable functions is measurable
QUESTION [5 upvotes]: I'm having trouble showing the following:

If $f_n$ is a sequence of measurable functions such that $f_n$ converges to $f$ almost everywhere, then $f$ is measurable.

I was thinking of using $\limsup$ since I know that $\limsup f_n$ is measurable. But now I'm not sure how to continue my argument.

REPLY [2 votes]: It is not always true.
See Exercise V chapter 3 of Bartle's Book.