A Radon-Nikodym theorem for monotone measures

TL;DR

This paper establishes a Radon-Nikodym theorem for the Choquet integral with respect to monotone measures, providing necessary and sufficient conditions for the existence and uniqueness of derivatives without prior assumptions.

Contribution

It introduces a Radon-Nikodym theorem for monotone measures, extending classical results to non-additive measures with minimal conditions.

Findings

Necessary and sufficient conditions for Radon-Nikodym property

Uniqueness of the Radon-Nikodym derivative under null-continuity

Radon-Nikodym type theorem for sigma-finite monotone measures

Abstract

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair $(\mu, \nu)$ of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function $f$. If $\nu$ is null-continuous and weakly null-additive, then $f$ is uniquely determined almost everywhere by $\nu$ and thus is called the Radon-Nikodym derivative of $\mu$ w.r.t. $\nu$. For $\sigma$-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.