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.