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Math Help - Lebesgue Integral

  1. #1
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    Lebesgue Integral

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

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

    Prove that for each \epsilon>0, there exist 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}.
    Since |f_n|\leqslant g for all n, it suffices to find a set E_\varepsilon \subseteq X with m(E_\varepsilon)< \infty such that if F\subseteq X and F \cap E_\epsilon =\emptyset, then \int_F |g|^p dm < \varepsilon^p for all 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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