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

Source abstract

This paper is the first of a series of two papers where we develop a point-free Lebesgue integration theory for functions on $\sigma$-locales. This series describes the integral of localic general functions 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. In this first part, we present the theory for the case of simple functions, laying the groundwork to develop the theory for more general functions.