Thread: Converging measurable sequence

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...

Originally Posted by Unenlightened

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, . If you know that measurability is preserved by sup's and inf's of sequences then the result follows immediately.