On the validity of the Radon-Nikodym Theorem

TL;DR

This paper introduces a new formulation of the Radon-Nikodym theorem using the concept of weak localizability, providing a necessary and sufficient condition for Radon-Nikodym-type representations in measure theory.

Contribution

It defines weak localizability for measures and proves it as a key condition for the Radon-Nikodym theorem, using elementary proof techniques.

Findings

Weak localizability is necessary and sufficient for Radon-Nikodym representations.

Provides a constructive method for envelope functions of measurable functions.

Main results rely on elementary tools like Markov's inequality and monotone convergence.

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.