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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- Ehud Hrushovski
- Affiliation: Hebrew University of Jerusalem, Department of Mathematics, Jerusalem, Israel
- Ya’acov Peterzil
- Affiliation: University of Haifa, Department of Mathematics and Computer Science, Haifa, Israel
- Anand Pillay
- Affiliation: University of Illinois, Department of Mathematics, Altgeld Hall, 1409 W Green Street Urbana, IL 61801, and University of Leeds, School of Mathematics, Leeds, LS2 9JT England
- MR Author ID: 139610
- Received by editor(s): July 16, 2006
- Published electronically: February 2, 2007
- Additional Notes: The first author was supported by the Israel Science Foundation grant no. 244/03
The last author was supported by NSF grants DMS-0300639 and FRG DMS-0100979, as well as a Marie Curie chair
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- 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
- MathSciNet review: 2373360