The idempotent Radon--Nikodym theorem has a converse statement

TL;DR

This paper characterizes which $\sigma$-maxitive measures possess the Radon--Nikodym property, providing a converse to the existing theorem and deepening understanding of idempotent integration's foundational aspects.

Contribution

It establishes a converse characterization of $\sigma$-maxitive measures with the Radon--Nikodym property, extending the classical theory of idempotent integration.

Findings

Identifies conditions under which $\sigma$-maxitive measures have the Radon--Nikodym property

Provides a converse theorem to Sugeno and Murofushi's Radon--Nikodym result

Enhances theoretical understanding of idempotent measure theory

Abstract

Idempotent integration is an analogue of the Lebesgue integration where $\sigma$-additive measures are replaced by $\sigma$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $\sigma$-maxitive measures that have the Radon--Nikodym property.