Let (fn) be a sequence of measurable functions that converges pointwise to a function f. Show that f is measurable.
It seems to me that this should be really easy - but I can't find the proof anywhere...
This is best done in terms of lim sup or lim inf. For example, $\displaystyle f(x) = \mathop{\sup}_{n\to\infty}\Bigl(\,\mathop{\inf}_{k \geqslant n}f_k(x)\Bigr)$. If you know that measurability is preserved by sup's and inf's of sequences then the result follows immediately.