Borel and Continuous Systems of Measures

TL;DR

This paper provides a detailed, measure-theoretic investigation of Borel and continuous systems of measures, focusing on their properties and mapping behaviors, to support categorical constructions in measured and topological groupoids.

Contribution

It offers a self-contained, thorough analysis of systems of measures, enhancing the theoretical foundation for categorical constructions involving groupoids with Haar systems.

Findings

Develops a comprehensive measure-theoretic framework for systems of measures

Clarifies properties of compositions, liftings, and disintegrations of measures

Supports future categorical and topological applications

Abstract

We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.