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Convergence (In Measure)

  1. Oct 9, 2008 #1
    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?
     
  2. jcsd
  3. Oct 10, 2008 #2
    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.
     
  4. Oct 10, 2008 #3

    morphism

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

    This is not the same as convergence in [itex]L^\infty[/itex].

    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".
     
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