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Homework Help: Measure Theory / Series of functions

  1. Dec 1, 2011 #1
    1. The problem statement, all variables and given/known data

    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.
  2. jcsd
  3. Dec 1, 2011 #2
    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]??
  4. Dec 1, 2011 #3
    I assume this is something that I won't be able to grasp within an hour...
  5. Dec 1, 2011 #4
    Will letting [tex]g_n(x)=-\frac{1}{n}[/tex] lead anywhere?
  6. Dec 1, 2011 #5

    Do you know a function that converges pointswize to 0, but whose integrals don't converge??
  7. Dec 1, 2011 #6
    I don't know, my brain is fried.

    [tex]f_n(x)=\frac{x}{n}\, ?[/tex]
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