# A problem on almost singular measures in real analysis

Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$ or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such that $\mu(E) > 0$ and $\nu \ge \epsilon \mu$ on $E$ (that is, every measurable set $S \subseteq E$ has $\nu(S) \ge \epsilon \mu(S)$).