The discussion confirms that if a sequence of measurable functions converges in measure, it is indeed possible to find a subsequence that converges almost uniformly across all measure spaces. This conclusion is supported by the F. Riesz convergence theorem, which establishes the relationship between convergence in measure and almost uniform convergence. The essential supremum of the difference between the functions approaches zero, but this is distinct from convergence in L^\infty.
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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.jostpuur said:Does the almost uniform convergence mean that the essential supremum of f-f_n approaches zero?