Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

L^1 convergence and uniform convergence

  1. Sep 10, 2009 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Let (X,µ) be a measure space with µ(X) infinite. I'm trying to find an example of a pair (X,µ) and a sequence {f_n} of L^1(X,µ) functions converging uniformly to a function f such that we do not have f_n --> f in L^1(X,µ).
     
  2. jcsd
  3. Sep 10, 2009 #2
    If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.
     
  4. Sep 11, 2009 #3

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    I don't see how you can conclude the statement is false from this.

    In fact it is possible to come up with an example satisfying the OP using X=R with Lebesgue measure. Take f_n to be a sequence of rectangles with decreasing height but increasing width, in such a way that f_n -> 0 uniformly but not in L^1. You can even rig it so that f_n doesn't converge in any L^p (1 <= p < inf).
     
  5. Sep 11, 2009 #4
    Doesn't L^1 mean the space of integrable functions? So why isn't f=0 in this space?
     
  6. Sep 11, 2009 #5
    Yes, f(x) = 0 is in L^1. The issue is whether f_n converges to f in the L^1 norm.
     
  7. Sep 11, 2009 #6

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Ah! Thank you morphism.
     
  8. Sep 11, 2009 #7
    What is the L^1 norm?
     
  9. Sep 11, 2009 #8

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The L^1 norm is defined on the space of all integrable functions as
    [tex]\| f\|_{L^1(X,\mu)}=\int_X |f|d\mu
    [/tex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: L^1 convergence and uniform convergence
  1. Uniform convergence (Replies: 4)

  2. Uniform convergence (Replies: 6)

  3. Uniform convergence (Replies: 6)

Loading...