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Math Help - A.e Convergent Problem.

  1. #1
    Junior Member
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    A.e Convergent Problem.

    Suppose i have \sum_{n=1}^{\infty} ||f_n||_2 < \infty. How to show that the \sum_{n=1}^{\infty} f_n converges absoutely almost everywhere ,
    f= \sum_{n=1}^{\infty} f_n \in L^2 , and ||f||_2 <= \sum_{n=1}^{\infty}||f_n||_2 .
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  2. #2
    Super Member
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    Re: A.e Convergent Problem.

    Quote Originally Posted by younhock View Post
    Suppose i have \sum_{n=1}^{\infty} ||f_n||_2 < \infty. How to show that the \sum_{n=1}^{\infty} f_n converges absoutely almost everywhere ,
    f= \sum_{n=1}^{\infty} f_n \in L^2 , and ||f||_2 <= \sum_{n=1}^{\infty}||f_n||_2 .
    All of this is a consequence of Minkowski's integral inequality, we have ( dn is the counting measure on \mathbb{N}):

    \sum \| f_n \|_2 = \int_{\mathbb{N}} \left( \int_X |f(n,x)|^2dx \right)^{1/2} dn \geq \left( \int_X \left( \int_{\mathbb{N}} |f(n,x)|dn \right)^2 dx \right)^{1/2} = \left( \int_X \left( \sum |f_n(x)| \right)^2 dx \right)^{1/2}\geq \| f\|_2

    Since the left hand side is finite, so is the right side, but an integrable function can only be infinite on a null set so we get that \left( \sum |f_n| \right)^2 is finite a.e. The rest follows directly from the inequality above as well.
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