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

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Tao, Terence

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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).

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Polish Acad Sciences Inst Mathematics-Impan

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Mathematics

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Fundamenta Mathematicae

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10.4064/fm226-7-2022

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