Generically stable and smooth measures in NIP theories

We formulate the measure analogue of generically stable types in first order theories with NIP (without the independence property), giving several characterizations, answering some questions from [9], and giving another treatment of uniqueness results from [9]. We introduce a notion of “generic compact domination”, relating it to stationarity of Keisler measures, and also giving group versions. We also prove the “approximate definability” of arbitrary Borel probability measures on definable sets in the real and p-adic fields.

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