On the validity of the Radon-Nikodym Theorem
General Mathematics
2025-12-03 v3
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.
Keywords
Cite
@article{arxiv.2506.12023,
title = {On the validity of the Radon-Nikodym Theorem},
author = {Paolo Roselli and Michel Willem},
journal= {arXiv preprint arXiv:2506.12023},
year = {2025}
}