# Lebesgue Integral: Practice Problem 2

1. Oct 15, 2014

### nateHI

1. The problem statement, all variables and given/known data
Let $f:\mathbb{R}\to \mathbb{R}$ be a nonnegative Lebesgue measurable function. Show that:
$lim_{n\to\infty}\int_{[-n,n]}f d\lambda=\int_{\mathbb{R}}f d\lambda$

2. Relevant equations

3. The attempt at a solution
Let $E_n=\{x:-n<x<n\}$ then write $f_n=f\mathcal{X}_{E_n}$
Now apply the Monotone Convergence Thm
$lim_{n\to\infty}\int_{[-n,n]} f d\lambda=lim_{n\to\infty}\int f_n d\lambda=\int_{\mathbb{R}}f d\lambda$

This seems correct. It's important I get it right though since it may be on the test. Please let me know if I did it correctly.

2. Oct 20, 2014

### Staff: Admin

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Oct 20, 2014

### Dick

Yes, it's right. But you have to at least say a word about why the premises of the Monotone Convergence Theorem are satisfied. If you do that you don't even have to ask whether it's right. You'll know.

4. Oct 21, 2014

### nateHI

EDIT: Deleted my reply. The test was so hard that he decided to make it a take home exam at the last minute. I'll repost my response after the due date.

Last edited: Oct 21, 2014
5. Oct 24, 2014

### nateHI

$E_n\subset E_{n+1} \implies f_n$is a monotone increasing sequence so $f_n<f_{n+1}$
from $f_n=f\mathcal{X}_{E_n}$ is is clear that $f_n\to f$

Unfortunately the test wasn't anywhere near as easy as the practice problems..lol

Last edited: Oct 24, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted