Riesz spaces of signed charges on semi-rings

TL;DR

This paper develops a constructive approach to defining supremums in Riesz spaces of signed measures on semi-rings, enabling Jordan decomposition for signed charges regardless of Riesz space structure.

Contribution

It introduces a constructive method for supremums in Riesz spaces of signed charges on semi-rings and applies it to establish Jordan decomposition.

Findings

Constructive supremum definition for set functions.

Jordan decomposition for signed charges on semi-rings.

Applicability regardless of Riesz space structure.

Abstract

A constructive definition of the supremum of a family of set functions is exploited in the context of Riesz spaces of signed measures and finitely additive functions (signed charges) on semi-rings. We explore applications, particularly to establish a Jordan decomposition for signed charges on semi-rings, whether the structure of Riesz space is present or not.