Could you, please, check the following:
Let's consider a real line $\mathbb{R}$ and let's define two measures on it. For some (possibly finite) sequences $s_{1}, s_{2}, \dots$ and $a_{1}, a_{2}, \dots$, s.t. $a_{i}>0$ and $\sum_{i}a_{i} = 1$, let $$ \begin{equation} \delta_{s_i}(X)=\begin{cases} 1, & \text{if $s_{i}\in X$}\\ 0, & \text{if $s_{i}\not\in X$} \end{cases} \end{equation} $$ for any Lebesgue measurable set $X$. Next, we define $$ \nu = \sum_{i}a_{i}\delta_{s_i}, $$ and $$ \mu = \sum_{i}\delta_{s_i}. $$
Then, one can see that $\nu$ is a discrete probability distribution on $s_{1}, s_{2}, \dots$.
Statement: $\nu$ is absolutely continuous wrt to $\mu$, also for any measurable set $A$ $$ \nu(A) = \int_{A} f d\mu, $$ where $f$ is Radon-Nikodym derivative and $$ \begin{equation} f(x)=\begin{cases} a_{i}, & \text{if $x=s_{i}$, for some $i$}\\ 0, & \text{otherwise}. \end{cases} \end{equation} $$