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Homework Help: Sigma-finite measures

  1. Mar 6, 2006 #1
    hello, I need some help with 2 problesm involving sigma-finite measures.

    *note* a set X is sigma-finite if X can be written as a compact union of subsets, i.e. X = Union of Xi for i = 1,2,3,...n, and the measure of each Xi is finite.

    q1)Given a measure space (X, M, m), let f be lebesgue measurable such that f(x)>0 for every x element of X, show that m is sigma-finite.
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    I think I may have solved this. Please see if I did anything wrong.

    Let X = union of arbitrary disjoint subsets Xi where i =1,2,3...n

    since f is lebesgue measurable, then f(x)m(X) is finite, hence m(X) is finite.
    Since m(X) = m (UXi) = summation m(Xi) is finite. Hence m(Xi) is finite.
    hence m is sigma finite. Is this correct?

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    q2) suppose f is sigma-finite, constrauct a lebesgue measurable function s.t. f(x) > 0 for every x element of X abd the integral of f(x)dx over X = 1.

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    I'm thinking this is how to solve this.
    since the integral of f(x)dx over X = 1 then
    f(x)m(X) = 1
    and
    f(x) = 1/m(X)...this f(x) is finite, hence it is lebesgue meaurable.

    I'm thinking I need to get rid of the m(x) part, but I have no idea how. help please?
     
  2. jcsd
  3. Mar 7, 2006 #2
    anyone? help please. thanks!
     
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