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haljordan45
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Homework Statement
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.
Homework 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
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.
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