Bochner integrals in ordered vector spaces

TL;DR

This paper introduces a new approach to extend Bochner integrability to functions valued in Archimedean directed ordered vector spaces, creating a structured space of integrable functions with order-preserving integration.

Contribution

It develops a method to cover Archimedean directed ordered vector spaces with Banach spaces and extends Bochner integrability to these spaces, preserving order.

Findings

Constructs a Banach space cover for Archimedean directed ordered vector spaces.

Extends Bochner integrability to functions with values in these spaces.

Ensures the integral is an order-preserving map.

Abstract

We present a natural way to cover an Archimedean directed ordered vector space $E$ by Banach spaces and extend the notion of Bochner integrability to functions with values in $E$. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.