Consider two singular measures $ m$ and $v$ and $v$ is absolutely continuous with respect to $m$ show that $v=0$ Consider two singular measures $m$ and $v$ on a measurable space $(X,\mathcal{A})$ and $v$ is absolutely continuous with respect to $m$, i.e., $(v<<m)$.  Show that $v=0$.

$m$ and $v$ are singular means there exist disjoint $A$ and $B$ such that $A\cup B = X$ and $m(A) = 0, v(B) = 0$.
Then for any $C \in \mathcal{A}$, we have
$$v(C) = v(C\cap A) + v (C\cap B) = v(C\cap A)$$
then since $m(C\cap A) =0$ and $v$ is absolutely continuous with respect to $m$, we have $v(C\cap A) =0$
