Syntactic Radon–Nikodym conjecture

Let $S_n$ be a syntactic fractal space with a family of definable sets $cF_n$, and let $\nu_n$ and $\mu_n$ be $S_n$-measures. Write $\nu_n \ll \mu_n$ when $\mu_n(A)=0$ implies $\nu_n(A)=0$ for every $A\in\mathcal{F}_n$. Let $L^1(S_n)$ denote the space of $\mathcal{F}_n$-definable functions integrable with respect to $\mu_n$. Syntactic Radon–Nikodym conjecture. If $\nu_n \ll \mu_n$ and both measures are $\Sigma^0_2(\mathcal{F}_n)$-definable, then there exists a function $f\in L^1(S_n)$ such that

$$\nu_n(A)=\int_A f(x)\,d\mu_n(x)$$

for all $A\in\mathcal{F}_n$. This is a constructive, definability-restricted analogue of the classical Radon–Nikodym theorem; the source provides no resolution, so the conjecture remains open.