Let $\mu_1$ and $\mu_2$ be finite positive measures on $(X,\Sigma)$. Show that there exist disjoint measurable sets $A\cup B=X$ such that $\mu_1 \bot \mu_2$ on $(A,\Sigma \cap A)$ and $\mu_1 \ll \mu_2 \ll \mu_1 $ on $(B,\Sigma \cap B)$.
Hint: show that $\mu_1 , \mu_2 \ll \mu_1 + \mu_2 $ and apply Radon-Nikodym theorem.
I couldn't solve this question. I'm certain it's been asked before but I don't know how to look for it. any hints or links would be appreciated.