Integrals for functions with values in a partially ordered vector space

TL;DR

This paper explores the integration of functions valued in partially ordered vector spaces and introduces two extension methods that generalize classical integrable function spaces.

Contribution

It proposes new extension frameworks for integrating functions in partially ordered vector spaces, broadening the scope of classical integration theory.

Findings

Two extension methods for integrable functions are developed.

Extensions recover the classical space of integrable functions for real-valued simple functions.

The approach generalizes classical integration to more abstract vector spaces.

Abstract

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.