Let $m$ be the Lebesgue measure and $v$ be the counting measure on the interval $[0,1]$. I know that the Lebesgue decomposition theorem does not hold true for $v$ wrt $m$ on this interval in the sense that $v$ cannot be split as follows:

$$
\nu=\nu\_a+\nu\_s,\space\space\nu\_a(E)=\int\_E(fdm),\space\space 
\nu\_s(X-X\_s)=m(X\_s)=0\space\space\space
$$

My problem is showing that the Lebesgue measure has no decomposition wrt $v$ in this same sense. I'm under the impression that I just need to pick a subset of $[0,1]$ that isn't Lebesgue measurable but that doesn't seem to be working. Any thoughts?