On the validity of the Radon-Nikodym Theorem
Paolo Roselli, Michel Willem
TL;DR
The paper extends Radon-Nikodym theory beyond σ-finiteness by introducing weakly localizable measures and a compatibility framework. It employs a variational approach and elementary tools to construct upper envelope densities and establish a local RN theorem, culminating in a global equivalence: μ is weakly localizable if and only if every compatible pair (μ,ν) satisfies the Radon-Nikodym property. The results provide a constructive pathway to RN densities via envelope functions and yield insight into the structural role of localizability in measure theory. This broadens the applicability of RN-type representations and clarifies the interplay between measure compatibility and density existence.
Abstract
This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure.