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

Abstract

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. $$\sum _{A\in {\pi }_n:\mu (A)>0}{1}_A\frac{\nu (A)}{\mu (A)}⟶\frac{d{\nu }^a}{d\mu }\mu \text{ a.e. as }n\to \infty .$$ 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.