Does Convergence in Measure Always Guarantee Almost Uniform Convergence?

[Thorn]

If a sequence of measurable functions (real-valued) converges in measure, is it true that you can find a subsequence that converges almost uniformly? (This is obviously true if m*(domain) is finite...but in general is it?) If so, can someone outline a little why?

 

[jostpuur]

Does the almost uniform convergence mean that the essential supremum of f-f_n approaches zero?

If so, I think I came up with a very simple counter example to your claim that m*(domain)<oo would be enough for this.

 

[morphism]

Does the almost uniform convergence mean that the essential supremum of f-f_n approaches zero?

No, the usual definition goes something like this: $f_n \to f$ almost uniformly if for every $\epsilon > 0$ there is a set E of measure less than $\epsilon$ such that $f_n \to f$ uniformly on the complement of E.

This is not the same as convergence in $L^\infty$.

What the OP is asking turns out to be true, for all measure spaces. It follows from a result that's sometimes called the "F. Riesz convergence lemma/theorem".