Riesz integral representation theory

TL;DR

This paper develops a Riesz integral representation theory for functions, operators, and measures valued in uniform commutative monoids, establishing a bijective correspondence between Riesz integrals and finitely additive Riesz measures.

Contribution

It introduces a new integral representation framework for operators in uniform commutative monoids, linking Riesz integrals with Riesz measures and characterizing operators with the Hammerstein property.

Findings

Every Riesz integral corresponds to a unique Riesz measure.

Operators with the Hammerstein property can be represented as Riesz integrals under certain conditions.

The theory applies to set-valued functions and excludes some spaces of infinitely differentiable functions.

Abstract

We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous). It describes the operators to which the theory can be applied and the finitely-additive measures they generate. Operators satisfying the conditions will be called ``Riesz integrals''. Given an underlying ``Riesz system'', it is shown that every Riesz integral generates a certain kind of finitely additive measure called here a ``Riesz measure''. The correspondence between Riesz integrals and Riesz measures is a bijection. A straightforward calculation shows that if an operator has such a representation, then it must have the Hammerstein property. For topological vector spaces, the theory yields necessary and sufficient conditions for operators with the Hammerstein property to be Riesz integrals. We note that uniform commutative monoids arise naturally when considering set-valued functions, and that the axioms of a Riesz system rule out certain spaces of infinitely differentiable functions.