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Math Help - Sequence in Lebesgue Space

  1. #1
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    Sequence in Lebesgue Space

    Show that if (f_n)_n is a sequence in L^1[0, 1] is such that \sum_{n=1}^{\infty}  || f_n ||_1 < \infty then \sum_{n=1}^{\infty} | f_n(s) | < \infty for almost every s \in [0, 1].


    This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.
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  2. #2
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    Quote Originally Posted by pascal4542 View Post
    Show that if (f_n)_n is a sequence in L^1[0, 1] is such that \sum_{n=1}^{\infty}  || f_n ||_1 < \infty then \sum_{n=1}^{\infty} | f_n(s) | < \infty for almost every s \in [0, 1].


    This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.
    You probably know that if f is nonnegative and \int f<\infty, then f(s)<\infty for almost every s (this is almost by definition of the integral).

    Then apply that to f=\sum_{n=1}^\infty |f_n(s)|... (You will need the monotone convergence theorem to justify the computation of \int f(=\int \lim_N \sum_{n=1}^N |f_n|)
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