# When does convergence in measure implies existence of subsequence converging almost everywhere

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.

• Yes, it's true in general. – carmichael561 May 6 '16 at 15:26