Lebesgue integration on $\sigma$-locales: simple functions

TL;DR

This paper develops a point-free approach to Lebesgue integration for simple functions on σ-locales, extending classical measure theory into a localic, point-free setting with new definitions for integrable functions.

Contribution

It introduces a novel point-free formulation of Lebesgue integration on σ-locales, generalizing measure and integrability concepts beyond Boolean algebra constraints.

Findings

Defines Lebesgue integral on σ-locales using measure on σ-sublocales

Extends integrability to localic general functions

Provides a framework for measure theory in a point-free setting

Abstract

This paper presents a point-free version of the Lebesgue integral for simple functions on $\sigma$-locales. It describes the integral with respect to a measure defined on the coframe of all $\sigma$-sublocales, moving beyond the constraints of Boolean algebras. It also extends the notion of integrable function, usually reserved for measurable functions, to localic general functions.