Measure Theory / Series of functions

[spitz]

Homework Statement

I am looking for an example of a series of funtions:
$$\sum g_n$$ on $$\Re$$

such that:

$$\int_{1}^{2}\displaystyle\sum_{n=1}^{\infty}g_n(x) \, dx \neq \displaystyle\sum_{n=1}^{\infty} \, \int_{1}^{2} \, g_n(x) \, dx$$

"dx" is the Lebesque measure.

2. The attempt at a solution

I haven't attempted a solution as I'm not sure how to approach this problem. If somebody could explain this to me or link to sample problems similar to this, I would really appreciate it.

 

[micromass]

Do you know the monotone convergence theorem?? Do you know counterexamples to the theorem when you don't assume that convergence is monotone?

That is: can you find a sequence of functions $(f_n)_n$ such that $f_n\rightarrow f$, but not $\int f_n\rightarrow \int f$??

 

[spitz]

I assume this is something that I won't be able to grasp within an hour...

 

[spitz]

Will letting $$g_n(x)=-\frac{1}{n}$$ lead anywhere?

 

[micromass]

Will letting $$g_n(x)=-\frac{1}{n}$$ lead anywhere?

No.

Do you know a function that converges pointswize to 0, but whose integrals don't converge??

 

[spitz]

I don't know, my brain is fried.

$$f_n(x)=\frac{x}{n}\, ?$$