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

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" set-ting 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 rep-resentation 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).