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Measure/Uniform Convergence

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data

    I would just like to be pointed in the right direction. I have this theorem:

    Let E be a measurable set of finite measure, and <fn> a sequence of measurable functions that converge to a real-valued function f a.e. on E. Then given ε>0 and [itex]\delta[/itex]>0, there is a set A[itex]\subset[/itex]E with mA<[itex]\delta[/itex], and an N such that for all x[itex]\notin[/itex]A and all n≥N,
    lfn(x)-f(x)l<ε

    To me it appears to be concluding:
    Given ε>0 and [itex]\delta[/itex]>0, there is a set A[itex]\subset[/itex]E with mA<[itex]\delta[/itex], and an N such that for all x[itex]\notin[/itex]A and all n≥N,
    <fn> converges uniformly to a real-valued function f on E~A.



    I know that this isn't case but I don't see why. So my question is what would the conclusion of this theorem need to say, in terms of ε and [itex]\delta[/itex], so that <fn> converges uniformly to a real-valued function f on E~A?


    Thank you for your time.
     
  2. jcsd
  3. Apr 17, 2012 #2
    The difference between definitions of uniform and pointwise convergence is really small. What you have there is really uniform convergence. Although I don't see why you need delta and A..
    Anyway, the difference is that in uniform convergence N does not depend on x. Swapping the conditions "there exists N" and "for all x" would produce the definition of pointwise convergence.
     
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