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Thread: Lebesgue Integral

  1. #1
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    Lebesgue Integral

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

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

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

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