Lebesgue Integral: Practice Problem 2

[nateHI]

Homework Statement

Let $f:\mathbb{R}\to \mathbb{R}$ be a nonnegative Lebesgue measurable function. Show that:
$lim_{n\to\infty}\int_{[-n,n]}f d\lambda=\int_{\mathbb{R}}f d\lambda$

Homework Equations

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.

 

[Greg Bernhardt]

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?

 

[Dick]

Homework Statement

Let $f:\mathbb{R}\to \mathbb{R}$ be a nonnegative Lebesgue measurable function. Show that:
$lim_{n\to\infty}\int_{[-n,n]}f d\lambda=\int_{\mathbb{R}}f d\lambda$

Homework Equations

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.

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.

 

[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.

 

[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