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What's the Difference Between

  1. Dec 1, 2007 #1
    1) Let [tex]E[/tex] be a measurable set of finite measure, and [tex]\{ f_n \}[/tex] a sequence of measurable functions that converge to a real-valued function [tex]f[/tex] a.e. on [tex]E[/tex]. Then, given [tex]\epsilon[/tex] and [tex]\delta[/tex], there is a set [tex]A\subset E[/tex] with [tex]mA < \delta[/tex], and an [tex]N[/tex] s.t. [tex]\forall x\notin A[/tex] and [tex]\forall n \ge N[/tex], [tex]|f_n(x) - f(x)| < \epsilon[/tex].

    2) Egorov's Theorem: Let [tex]E[/tex] be a measurable set of finite measure, and [tex]\{ f_n \}[/tex] a sequence of measurable functions that converge to a real-valued function [tex]f[/tex] a.e. on [tex]E[/tex]. Then there is a subset [tex]A\subset E[/tex] with [tex]mA < \delta[/tex] s.t. [tex]f_n[/tex] converges to [tex]f[/tex] uniformly on [tex]E\setminus A[/tex].

    Most texts prove #2 from #1, and I'm confused as to what the difference is. I always thought the definition of uniform convergence was that if [tex]\epsilon > 0[/tex] is given, we can choose an [tex]N[/tex] such that [tex]\forall n \ge N[/tex], [tex]|f_n(x)-f(x)| < \epsilon[/tex].

    Sorry if this is a stupid question, but I can't seem to wrap my brain around it. Thanks for the help.
     
  2. jcsd
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