A Radon-Nikod\'ym Theorem for Valuations

TL;DR

This paper investigates conditions under which one valuation has a density with respect to another, extending classical measure theory concepts to valuations and establishing a Radon-Nikodým type theorem.

Contribution

It establishes a Radon-Nikodým theorem for valuations, identifying necessary and sufficient conditions for the existence of a density between valuations.

Findings

Valuations must be absolutely continuous for a density to exist

A Hahn decomposition property is necessary and always holds for measures

The theorem extends classical measure theory to valuations

Abstract

We enquire under which conditions, given two $\sigma$-finite, $\omega$-continuous valuations $\nu$ and $\mu$, $\nu$ has density with respect to $\mu$. The answer is that $\nu$ has to be absolutely continuous with respect to $\mu$, plus a certain Hahn decomposition property, which happens to be always true for measures.