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We collect several foundational results regarding the interaction betweenlocally compact spaces, probability spaces and probability algebras, andcommutative $C^*$-algebras and von Neumann algebras equipped with traces, inthe "uncountable" setting in which no separability, metrizability, or standardBorel hypotheses are placed on these spaces and algebras. In particular, wereview the Gelfand dualities and Riesz representation theorems available inthis setting. We also present a canonical model that represents probabilityalgebras as compact Hausdorff probability spaces in a completely functorialfashion, and apply this model to obtain a canonical disintegration theorem andto readily construct various product measures. These tools are useful inapplications to "uncountable" ergodic theory (as demonstrated by the authorsand others).

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