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Math Help - Sequence, Lebesgue Space

  1. #1
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    Sequence, Lebesgue Space

    Show that if || f_n  ||_1 \leq 2^{-n} for every n \geq 1 then (f_n)_n converges to zero a.e.

    This should follow from

    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].


    However, I don't see how to show this. I would appreciate some hints on how to proceed.
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  2. #2
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    Quote Originally Posted by eskimo343 View Post
    Show that if || f_n  ||_1 \leq 2^{-n} for every n \geq 1 then (f_n)_n converges to zero a.e.

    This should follow from

    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].


    However, I don't see how to show this. I would appreciate some hints on how to proceed.
    Not true, take f_n(x)=1 if x=0 and f_n(x)=0 otherwise. Then \sum\parallel f_n\parallel_1<\infty but \sum|f_n(0)|=\infty
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  3. #3
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    Quote Originally Posted by putnam120 View Post
    Not true, take f_n(x)=1 if x=0 and f_n(x)=0 otherwise. Then \sum\parallel f_n\parallel_1<\infty but \sum|f_n(0)|=\infty
    That does not contradict the assertion that \sum_{n=1}^{\infty} | f_n(s) | < \infty for almost every s \in [0, 1].

    Quote Originally Posted by eskimo343 View Post
    Show that if || f_n  ||_1 \leq 2^{-n} for every n \geq 1 then (f_n)_n converges to zero a.e.
    Let g_N = \sum_{n=N+1}^\infty|f_n|. Then \|g_N\|_1\leqslant\sum_{n=N+1}^\infty\|f_n\|_1\leq  slant2^{-N} (sum of geometric series). Define S_N = \{x:g_N(x)\geqslant2^{-N/2}\}. Then \mu(S_N)\leqslant2^{-N/2}, where \mu denotes the measure.

    If x\notin S_N then |f_n(x)|<2^{-N/2} for all n>N. Therefore if x\notin\limsup_{N\to\infty}S_N then f_n(x)\to0 as n\to\infty. But \limsup_{N\to\infty}S_N is a null set.
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