One can formulate the Radon-Nikodym so that it applies to all measure spaces without any restriction. Let $(X,\Sigma,\nu)$ be a measure space and $\nu$ be a real valued signed measure on $\Sigma$.

Then $\nu$ is absolutely continuous with respect to $\mu$ if and only if for all $\epsilon>0$ there exists $\delta>0$ such that for all $A\in\Sigma$, $\mu(A)\leq\delta$ implies $|\nu(A)|\leq\epsilon$.

We say that $\nu$ is *truly continuous* with respect to $\mu$ if and only for all $\epsilon>0$ there exists $\delta>0$ and $F\in\Sigma$ with $\mu(F)<\infty$ such that for all $A\in\Sigma$, $\mu(A\cap F)\leq\delta$ implies $|\nu(A)|\leq\epsilon$.

If $(X,\Sigma,\nu)$ is $\sigma$-finite, then a real-valued signed measure $\nu$ on $\Sigma$ is truly continuous with respect of $\mu$ if and only if it is absolutely continuous with respect to $\mu$.

**Theorem:** Let $(X,\Sigma,\nu)$ be a measure space and $\nu$ be a real valued signed measure on $\Sigma$. Then $\nu$ is truly continuous with respect to $\mu$ if and only if there exists a measurable function $f:\Sigma\to \mathbb{R}$ such that $$\nu(A)=\int_A f~\text{d}\mu$$ for all $A\in\Sigma$.

This formulation of the Radon-Nikodym is from the second volume of David Fremlin's treatise on measure theory. Fremlin doesn't assume $\nu$ to be countably additive, which requires additional distinctions. There is also a treatment in Section 5.4 of the book Measure and Integration by Dietmar Salamon (preprint available here), which also contains a useful characterization by Heinz König of being truly continuous in terms of absolute continuity and an inner regularity property