Differentiation of measures on a non-separable space, and the Radon-Nikodym theorem

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$\sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)} \longrightarrow \frac{d\nu^a}{d\mu} \quad \mu \text{ a.e. as } n\to \infty .$$ As an application, we modify the probabilistic proof of the Radon-Nikodym Theorem so that it uses convergence along a properly chosen sequence (instead of along a net), and so that it does not rely on the martingale convergence theorem (nor any probability theory), obtaining a completely elementary proof.