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Generically stable and smooth measures in NIP theories

2012; American Mathematical Society; Volume: 365; Issue: 5 Linguagem: Inglês

10.1090/s0002-9947-2012-05626-1

1088-6850

Ehud Hrushovski, Anand Pillay, Pierre Simon,

Algebraic Geometry and Number Theory

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 an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.