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A Question on Lebesgue Decomposition

  1. Nov 21, 2013 #1
    1. The problem statement, all variables and given/known data

    Let μ be the counting measure and m be the Lebesgue measure. Then show that on the interval [0,1] m has no Lebesgue decomposition with respect to μ.

    2. Relevant equations

    If such a decomposition exists, then the following holds true where X is the whole space, E is a subset of X, and Xs is the singular subset of the space:

    1. m=ma+ms where ma is absolutely continuous and ms is singular
    2. ma(E)=∫Efdμ
    3. ms(X-Xs)=μ(Xs)=0

    3. The attempt at a solution

    I know how to show that μ has no Lebesgue decomposition with respect to m, but can't seem to figure out this direction. I'm assuming that I need to pick a set E that contradicts 3 above, but I'm at a loss.
     
    Last edited: Nov 21, 2013
  2. jcsd
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