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Let $f_{n}$ be a sequence of integrable functions on $R^d$ that converges to $f$ a.e., and assume there exists a number $c$ such that for all $n$, $\int|f_{n}(x)| <c$. Prove that $$\lim_{n \rightarrow \infty} \left( \int |f_n| - \int|f_n - f|\right) = \int |f|$$.

It looks to me like possibly the triangle inequality may be useful? But I am really not sure where to begin.

Jared Stewart
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  • It might be related to the Brezis-Lieb Lemma (http://anhngq.wordpress.com/tag/brezis-lieb-lemma/). However, I do not really understand the assumption that ${|f_n|_\infty}$ is a bounded sequence. – Siminore Oct 14 '13 at 17:06
  • Sorry, I omitted an integral sign! It should have read $\int |f_{n}|<c$. – Jared Stewart Oct 14 '13 at 17:18
  • The it is precisely that case $j(t)=|t|$ of the Brezis-Lieb Lemma. Its proof is not really trivial... – Siminore Oct 14 '13 at 17:21

1 Answers1

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  1. By Fatou's lemma, $f$ is integrable (and so is $f_n-f$ for each $n$).

  2. Define $g_n:=|f|+|f_n-f|-|f_n|$. Since $0\leqslant g_n\leqslant 2|f|$ and $g_n\to 0$ almost everywhere, we can conclude by dominated convergence.

Davide Giraudo
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