Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
11  
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
2  
Alex, maybe you'd care to type up an answer...? –  The Chaz 2.0 Dec 7 '11 at 17:20

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.