Groups, measures, and the NIP

Hrushovski, Ehud; Peterzil, Ya’acov; Pillay, Anand

doi:10.1090/S0894-0347-07-00558-9

Groups, measures, and the NIP

Authors: Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay
Journal: J. Amer. Math. Soc. 21 (2008), 563-596
MSC (2000): Primary 03C68, 03C45, 22C05, 28E05
DOI: https://doi.org/10.1090/S0894-0347-07-00558-9
Published electronically: February 2, 2007
MathSciNet review: 2373360
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Abstract: We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of “compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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