$||f_n-f||_1 \to 0 $ iff $ \int| f_n|\ to \int|f| $ if $f_n \to f $ a.e.

Asked Feb 17 '15 at 09:05

Active Feb 17 '15 at 09:05

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Assume $f_n , f \in L^1 $ and almost every where we have $ f_n \to f$ then I want to show that $\int|f_n-f| \to 0$ iff $\int|f_n| \to \int|f|$

One side is abvious by trinagle inequality , for the other side I think I must use a theorem like Dominated covergence theorem but I can not reach a conclusion. Is there a modiffied form of dominated covergence theorem.

I need some hint , thanks.

asked Feb 17 '15 at 09:05

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You can argue as in this post. – David Mitra Feb 17 '15 at 09:11
Use $|f_n - f| \leq |f_n| + |f|$. Your condition implies that $|f_n| + |f|$ is integrable. – Feb 17 '15 at 09:12

$||f_n-f||_1 \to 0 $ iff $ \int| f_n|\ to \int|f| $ if $f_n \to f $ a.e.

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