  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).
