Measure Theory / Series of functions

1. Dec 1, 2011

spitz

1. The problem statement, all variables and given/known data

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.

2. Dec 1, 2011

micromass

Staff Emeritus
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 $\int f_n\rightarrow \int f$??

3. Dec 1, 2011

spitz

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

4. Dec 1, 2011

spitz

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

5. Dec 1, 2011

micromass

Staff Emeritus
No.

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

6. Dec 1, 2011

spitz

I don't know, my brain is fried.

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

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