Borel measure absolutely continuous wrt Lebesgue integral

Question

Is there a non-zero Borel measure $\mu$ on $\mathbb R$, absolutely continuous with respect to Lebesgue measure, such that $\mu (U)$ is integer for every open set $U$? Counterexample or Proof?

I have no idea how to approach.

Answer 1

If $\mu$ is absolutely continuous with respect to Lebesgue measure then $\mu (E)$ approaches 0 as Lebesgue measure of E approaches 0. Being integer valued it follows that all sufficiently small intervals have measure 0 under $\mu$. Hence $\mu$ is the zero measure.