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Measure Theory / Series of functions

  • Thread starter spitz
  • Start date
  • #1
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Homework Statement



I am looking for an example of a series of funtions:
[tex]\sum g_n[/tex] on [tex]\Re[/tex]

such that:

[tex]\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[/tex]

"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.
 

Answers and Replies

  • #2
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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 [itex](f_n)_n[/itex] such that [itex]f_n\rightarrow f[/itex], but not [itex]\int f_n\rightarrow \int f[/itex]??
 
  • #3
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I assume this is something that I won't be able to grasp within an hour...
 
  • #4
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Will letting [tex]g_n(x)=-\frac{1}{n}[/tex] lead anywhere?
 
  • #5
22,097
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Will letting [tex]g_n(x)=-\frac{1}{n}[/tex] lead anywhere?
No.

Do you know a function that converges pointswize to 0, but whose integrals don't converge??
 
  • #6
60
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I don't know, my brain is fried.

[tex]f_n(x)=\frac{x}{n}\, ?[/tex]
 
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