Let $\nu$ and $\mu$ be positive $\sigma$-finite measures on $(S,\Sigma)$ with $\nu \ll \mu$ and let $h = \frac{d\nu}{d\mu}$.
I want to show that \begin{align} \mu\big(\{h=0\}\big)=0 \iff \mu \ll \nu. \end{align} Therefore, in the "$\impliedby$" direction, I assume that $\mu\big(\{h=0\}\big)>0$ but I face some difficulties with finding an exhausting sequence $(A_k)_{k \in \mathbb{N}} \subset S$ for both $\mu$ and $\nu$.
Beside this, in the "$\implies$" direction, how to approach this problem?
Suppose $\mu(h = 0) = 0$. Let $A$ be such that $\nu(A) = 0$. Then $\int_A h\ d\mu = 0$. Then $\int_{A\cap\{h > 0\}} h\ d\mu = 0$. Then $\mu(A\cap\{h>0\}) = 0$. Then $\mu(A) = 0$. Then $\mu \ll \nu$.
Conversely, suppose that $\mu \ll \nu$. Then, since $\nu(\{h = 0\}) = \int_{\{h = 0\}} h\ d\mu = 0$, $\mu(\{h = 0\}) = 0$.