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On a finite measure space, if $f_n \rightarrow f$ in measure as $n \rightarrow \infty$, then there is a subsequence $f_{n_k}$ of $f_n$ such that $f_{n_k} \rightarrow f$ almost everywhere as $k \rightarrow \infty$. Is this still true for non-finite measure spaces? The proof that I have doesn't seem to rely on finiteness, and I can't think of any counterexample.

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  • $\begingroup$ Yes, it's true in general. $\endgroup$ – carmichael561 May 6 '16 at 15:26
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This does hold for infinite measure spaces, see for instance Bass (http://bass.math.uconn.edu/real.html) Chapter 10.

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