1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    No.

    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]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Measure Theory / Series of functions
  1. Measure theory (Replies: 7)

  2. Measure theory (Replies: 1)

  3. Measure theory (Replies: 2)

  4. Measure theory (Replies: 4)

Loading...