Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration

TL;DR

This paper explores foundational aspects of uncountable measure theory, including dualities, representation theorems, and canonical models, to facilitate advanced analysis in ergodic theory without standard separability assumptions.

Contribution

It introduces a canonical model for probability algebras as compact Hausdorff spaces and develops disintegration and product measure constructions in the uncountable setting.

Findings

Established Gelfand dualities in the uncountable setting

Presented a functorial canonical model for probability algebras

Derived a canonical disintegration theorem and product measure constructions

Abstract

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools are useful in applications to "uncountable" ergodic theory (as demonstrated by the authors and others).