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Math Help - Lebesgue Integral

  1. #1
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    Lebesgue Integral

    Suppose g\in L_p(X) and |f_n| \le g,show that for each \epsilon >0, there is a set E_\epsilon \subseteq X with m(E_\epsilon) < \infty such that if F \subseteq X and F\cap E_\epsilon = \phi ,then

    \int_F |f_n|^p dm <\epsilon^p, for all n\in \mathbb{N}
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  2. #2
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    Quote Originally Posted by problem View Post
    Suppose g\in L_p(X) and |f_n| \le g,show that for each \epsilon >0, there is a set E_\epsilon \subseteq X with m(E_\epsilon) < \infty such that if F \subseteq X and F\cap E_\epsilon = \phi ,then

    \int_F |f_n|^p dm <\epsilon^p, for all n\in \mathbb{N}
    Note that it suffices to prove the bound for g, since \int |f_n|^p dm\leq \int |g|^p dm.

    I guess X is supposed to be \sigma-finite: there is an increasing sequence (E_k)_{k\geq 0} of measurable subsets with \mu(E_k)<\infty and X=\bigcup_k E_k. Then you should justify that \int_{E_k} |g|^p dm\to \int |g|^p dm and use this to conclude.
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