Here is a problem I want to confirm my answer to.
Let $\mu $ and $\nu$ be two non-negative measures on a $\sigma$ - algebra $M$. If $\mu$ is mutually singular with $\nu$ and absolutely continuous with respect to $\nu$, show that $\mu=0$
My proof: Since $\mu$ is mutually singular with $\nu$, that implies the support of $\mu$ and the support of $\nu$ have an empty intersection. Now, if $\mu$ is not identically $0$, there exists an $A \in M$ such that $\mu(A)>0.$ But then $\nu(A)>0$ as well since, if $\nu(A)=0$, then $\mu(A)=0$. But then $A$ is a set for which both measures are positive, i.e. $A$ is in both the support of $\mu$ and the support of $\nu$. But that's a contradiction since support of $\mu$ and support of $\nu$ have empty intersection.
Thanks!