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!

Homework Help: Measurable with Respect to Complete Space

  1. Dec 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Let f:(X,A,μ)->[0,infinity] have a Lebesgue integral, meaning that the inf(upper lebesgue sum)=sup(lower lebesgue sum)=L for a finite L. Show that f is measurable with respect to the completion of the sigma algebra A with respect to μ. You may fix an integrable set E.

    2. Relevant equations

    Upper Lebesgue Sum(P,f)=Ʃ(supf)μ(Ej)
    Lower Lebesgue Sum(P,f)=Ʃ(inf f)μ(Ej)

    3. The attempt at a solution

    What I have so far is the existence of simple functions un with un≤f. I'm assuming that un is an increasing sequence that approaches some limit u. Applying the Monotone Convergence Theorem lets me say that ∫u≤∫f, but I'm not sure what to do from here. I feel like showing that u=f almost everywhere may be enough to prove the measurability of f, but I don't know how to argue that it's true. Any help would be great.

    Also, sorry if it's a little difficult to read. I'm new to physicsforums and haven't quite figured out how to post using Tex commands just yet.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted