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Math Help - Converging measurable sequence

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    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...
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    Quote Originally Posted by Unenlightened View Post
    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, 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.
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