On NIP and invariant measures

Abstract

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 [16]. Among key results are (i) if $p=\mathrm{tp}(b/A)$ 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 $\mathrm{bdd}(A)$, (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.