Let $(f_n)$ be a sequence of measurable functions that converges almost everywhere to a measurable function $f$. Assume that there is an integrable function $g$ such that $|f_n|\leq g$ for all $n$ almost everywhere. Show that $(f_n)$ converges almost uniformly to $f$.
Now I don't know of any other sufficient condition for almost uniform convergence other that Ergorov's theorem. However, to apply it I need my space to be of finite measure, and somehow the existence of a dominating integrable function $g$ has to play a role. But I wouldn't know where to start, so I'd like some hint on how to start working on the problem. Thank you in advance!