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Math Help - Uniform Convergence

  1. #1
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    Uniform Convergence

    Let E=[0,2] and f_{n}= \chi_{[1/n,2/n]}, \ n=1,2,.... Show that f_{n} converges almost uniformly on E but not uniformly.
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  2. #2
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    To see convergence is not uniform, observe that for every 0<\epsilon<1, no matter how large an n you pick, |f_n|=1>\epsilon on [1/n,2/n].

    To see convergence is almost uniform, observe that the set above is the only set on which uniform convergence fails. So let \epsilon>0 and chuse n so large that \mu([1/n,2/n]) =: \mu(B) < \epsilon. Then f_n restricted to [0,2]\B is identically zero, which clearly converges uniformly to 0.
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