# A Question on Lebesgue Decomposition

1. Nov 21, 2013

### haljordan45

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