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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
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