# Showing that $\int \liminf_{n \to \infty} f_{n}(x) dx \le \liminf_{n \to \infty} \int f_{n}(x) dx$

I don't know why, but I find this problem counter intuitive to me.

Prove that if $\{f_n\}$ is a sequence of measurable nonnegative functions on a measurable set $E$ and $f(x)=\liminf_{n \to \infty} f_{n}(x)$, then

$$\int_{E} f(x) dx \le \liminf_{n \to \infty} \int_{E} f_{n}(x) dx.$$

Can someone outline the proof for me please?

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Fatou's lemma - proof is givein in Wikipedia's article: en.wikipedia.org/wiki/… –  Martin Sleziak Dec 7 '11 at 16:31
haha, how stupid i am. Thx. –  Alex J. Dec 7 '11 at 16:41
Alex, maybe you'd care to type up an answer...? –  The Chaz 2.0 Dec 7 '11 at 17:20
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## 1 Answer

This is called Fatou's lemma. You can also read the proof in Terry Tao's lecture notes. (See Corollary 16 on that page.) There is also a very nice remark for this lemma there.

For me, the key point is the definition of $\liminf$.

I am a little surprised that the book in which you learn measure theory does not mention the name of this statement.

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