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Math Help - Lebesgue Integral

  1. #1
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    Lebesgue Integral

    Let the sequence of integrable functions ( f_n ) be defined on the interval [0,\infty) in which f_n = \dfrac{1}{n} 1_{[0,n]} on [0,\infty).
    Prove that there is no integrable function that dominates the sequence.

    My work:
    Suppose there exist a function  f that dominates the sequence.
    Hence,
    \int_{[0,\infty)} f dm
    >\int_{[0,n]} f dm
    >\sum_{k=1}^{n} \int_{[k-1,k)} f dm
    >\sum_{k=1}^{n} \int_{[k-1,k)} \dfrac{1}{n} dm

    I was stuck here.
    Can anyone help me to proceed?
    Or if this is not the way to prove,can anyone please show me the correct way?
    Thanks.
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  2. #2
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    If there existed such a dominating function, you could apply Lebesgue Dominated Convergence theorem, but

    \lim_nf_n=0 pointwise (even uniformly!)

    and

    \int f_n=1 for each n

    Hence we can't interchange limit and integral, a contradiction with having the condictions to apply Lebesgue theorem.
    Last edited by Enrique2; November 9th 2009 at 04:24 AM. Reason: correcting formula
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