What is the role of continuity in continuous linear forms representation?

TL;DR

This paper explores the role of continuity in the representation of continuous linear forms within domain theory, extending classical measure and linear form concepts to idempotent and lattice-valued contexts.

Contribution

It provides new representation theorems for continuous linear forms on domain-valued modules and applies these to idempotent Radon--Nikodym and Riesz theorems.

Findings

Representation theorems for continuous linear forms

Extension of Radon--Nikodym theorem to idempotent measures

Extension of Riesz representation theorem in domain-theoretic setting

Abstract

The recent extensions of domain theory have proved particularly efficient to study lattice-valued maxitive measures, when the target lattice is continuous. Maxitive measures are defined analogously to classical measures with the supremum operation in place of the addition. Building further on the links between domain theory and idempotent analysis highlighted by Lawson (2004), we investigate the concept of domain-valued linear forms on an idempotent (semi)module. In addition to proving representation theorems for continuous linear forms, we address two applications: the idempotent Radon--Nikodym theorem and the idempotent Riesz representation theorem. To unify similar results from different mathematical areas, our analysis is carried out in the general Z framework of domain theory.