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Thread: Lebesgue Integral

  1. #1
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    Lebesgue Integral

    Let $\displaystyle (f_n) \in L^p(X)$ for all $\displaystyle n \in \mathbb{N}$ and $\displaystyle 1\le p <\infty $.Suppose there exist a function $\displaystyle g \in L^p(X)$ such that $\displaystyle |f_n| \le g $for all $\displaystyle n \in \mathbb{N}$.

    Prove that for each $\displaystyle \epsilon>0$, there exist 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
    Let $\displaystyle (f_n) \in L^p(X)$ for all $\displaystyle n \in \mathbb{N}$ and $\displaystyle 1\le p <\infty $.Suppose there exist a function $\displaystyle g \in L^p(X)$ such that $\displaystyle |f_n| \le g $for all $\displaystyle n \in \mathbb{N}$.

    Prove that for each $\displaystyle \epsilon>0$, there exist 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 $\displaystyle n \in \mathbb{N}.$
    Since $\displaystyle |f_n|\leqslant g$ for all n, it suffices to find a set $\displaystyle E_\varepsilon \subseteq X$ with $\displaystyle m(E_\varepsilon)< \infty$ such that if $\displaystyle F\subseteq X$ and $\displaystyle F \cap E_\epsilon =\emptyset$, then $\displaystyle \int_F |g|^p dm < \varepsilon^p$ for all $\displaystyle n \in \mathbb{N}.$ That is a question that you have raised in this forum previously, and you'll find the answer here.
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