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I was wondering what is wrong in the following proof:

Proposition. Let $\{f_{n}\}$ be a sequence of integrable functions such that $f_{n}$ converges pointwise to a function $f$. Show that if:

$\lim \int |f_{n} - f| d\mu =0$ then $\int |f| d\mu = \lim \int |f_{n}| d\mu$.

Well I used the fact that $||f_{n}| - |f| | \leq |f_{n} - f|$ and integrating both sides and using the assumption that $\lim \int |f_{n} - f| d\mu =0$ I get the result. But I never used the fact that $f_{n}$ converges pointwise to $f$. Why do we need this assumption or what is wrong?

Thanks.

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    $f_n - f$ could be $0$ except on a "moving" set of measure ${1 \over n}$ where it's $\sqrt{n}$.2010-11-17
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    You are right that you don't need the assumption of pointwise convergence to prove this statement. You do need it to prove the _converse_; perhaps that is what was meant2010-11-17
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    @Qia.. Why is that? Not from Lebesgue dominated convergence theorem..2010-11-17
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    @TCL: actually it follows from (a clever application of) Fatou's lemma.2010-11-17

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