Categorical Geometry and Integration Without Points

TL;DR

This paper develops a point-free, categorical approach to measure theory and integration in infinite-dimensional spaces, extending classical concepts and establishing a functorial correspondence for finite-additive functions.

Contribution

It introduces a general point-free concept of measurability and proves that finite-additive functions can be uniquely extended to measures on abstract σ-algebras, inspired by categorical and point-free topology ideas.

Findings

Finite-additive functions extend to measures on abstract σ-algebras

Point-free measurability generalizes classical Lebesgue integration

Categorical approach characterizes Segal space by canonical data

Abstract

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article \cite{segal}. In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical $\sigma$-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function $\mu$ with values in $[0,1]$ can be extended to a measure on an abstract $\sigma$-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.