Journal of the European Mathematical Society

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Volume 13, Issue 4, 2011, pp. 1005–1061
DOI: 10.4171/JEMS/274

On NIP and invariant measures

Ehud Hrushovski[1] and Anand Pillay[2]

(1) Einstein Institute of Mathematics, Hebrew University, Givat Ram, 91 904, JERUSALEM, ISRAEL
(2) Department of Pure Mathematics, University of Leeds, LS2 9JT, LEEDS, UNITED KINGDOM

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of the paper \cite{NIP}. Among key results are (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$, (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of “generically stable" types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in $o$-minimal expansions of real closed fields.

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Hrushovski Ehud, Pillay Anand: On NIP and invariant measures. J. Eur. Math. Soc. 13 (2011), 1005-1061. doi: 10.4171/JEMS/274