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Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
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Locally compact Polish groups acting on standard Lebesgue spaces
I think that the analogue of a Bernoulli shift is rather the translation action on a Poisson point process. It appears in the work of Ornstein and Weiss on entropy theory of amenable groups as the ana …