On the functor of comonotonically maxitive functionals

TL;DR

This paper introduces a new functor of functionals that preserve maximum of comonotone functions, showing it is isomorphic to the capacity functor via the fuzzy max-plus integral, extending classical measure-function correspondences.

Contribution

It defines a novel functor of comonotonically maxitive functionals and proves its isomorphism to the capacity functor, extending Riesz's theorem to an idempotent setting.

Findings

The functor preserves maximum of comonotone functions and addition of constants.

It is a subfunctor of order-preserving functionals and contains the idempotent measure functor.

The functor is isomorphic to the capacity functor via the fuzzy max-plus integral.

Abstract

We introduce a functor of functionals which preserve maximum of comonotone functions and addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and contains the idempotent measure functor as subfunctor. The main aim of this paper is to show that this functor is isomorphic to the capacity functor. We establish such isomorphism using the fuzzy max-plus integral. In fact, we can consider this result as an idempotent analogue of Riesz Theorem about a correspondence between the set of $\sigma$-additive regular Borel measures and the set of linear positively defined functionals.