Consider a measurable space $\left( E, \mathcal{E}, \mu \right)$, where $\mu$ is $\Sigma$-finite, i.e., there is a sequence $(\mu_n)_{n=1}^{\infty}$ of finite measures on $\left( E, \mathcal{E} \right)$ such that $\mu = \sum_{n=1}^{\infty}\mu_n$. Define $\nu=\sum_{n=1}^{\infty}\dfrac{1}{\mu_n(E)2^n}\mu_n$. It is immediate to see that $\nu$ is a measure on $\left( E, \mathcal{E} \right)$, it is finite, and that $\mu \ll \nu \;$, i.e., $\; \nu(A)=0 \implies \mu(A)=0 \; \; \forall A \in \mathcal{E}$. Hence, by Radon-Nikodym theorem, $\exists ! \; p, \;$ $\mathcal{E}$-measurable and positive, such that $\forall f \in \mathcal{E}_+ \; \; \mu f = \nu (pf)$.
My attempt
The difficulties that I encounter have to do with the fact that apparently $p$ needs to vary with respect to $n$, that is, I cannot express $p$ in a way that works over the sum.
Trying to explicit the equivalence by Radon-Nikodym, it has to be
\begin{equation} \int_E\sum_{n=1}^{\infty}\mu_n(dx)f(x)=\int_E\sum_{n=1}^{\infty}\mu_n(dx)\dfrac{1}{\mu_n(E)2^n}p(x)f(x) \end{equation}
I start by taking $f$ to be an indicator function, i.e. $f = \mathbb{I}_A, \; A \in \mathcal{E}$ as I think that the rest would come more or less straightforward. Hence, the equivalence above becomes
\begin{equation} \sum_{n=1}^{\infty}\int_A\mu_n(dx)=\int_A\sum_{n=1}^{\infty}\mu_n(dx)=\int_A\sum_{n=1}^{\infty}\mu_n(dx)\dfrac{1}{\mu_n(E)2^n}p(x)= \\ \sum_{n=1}^{\infty}\dfrac{1}{\mu_n(E)2^n}\int_A \mu_n(dx)p(x) \end{equation}
(This post helped me take the sum out of the integral in the last equality) so, I try to work on
\begin{equation} \sum_{n=1}^{\infty}\int_A\mu_n(dx)=\sum_{n=1}^{\infty}\dfrac{1}{\mu_n(E)2^n}\int_A \mu_n(dx)p(x) \end{equation}
Of course, in the simple case where a sum is not involved, i.e., $\int_A \mu_n(dx) = \int_A \dfrac{1}{\mu_n(E)2^n} \mu_n(dx) p(x)$, it is easy to observe that $p(x)=\mu_n(E)2^n\mathbb{I}_E(x)$ is a solution and hence is the only solution (up to a negligible set). But when a sum is involved, even a sum with only two terms, I do not understand how to express $p$. Any help or hint on how to proceed would be more than appreciated.
