On NIP and invariant measures

On NIP and invariant measures

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 = 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...

Groups, measures, and the NIP

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...

Definable and Invariant Types in Enrichments of NIP Theories

On Invariant Measures

Given a measurable transformation on a measure space one can ask whether or not there is an equivalent measure that is invariant under the transformation. This problem is discussed very thoroughly in Halmos' Lectures on ergodic theory, pp. 81-90, 97. The first result along these lines is due to E. Hopf who obtained necessary and sufficient conditions for the existence of a finite invariant meas...

Generically stable and smooth measures in NIP theories

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 [9], and giving another treatment of uniqueness results from [9]. We introduce a notion of “generic compact domination”, relating it to stationarity of Keisler measures, and also giving group versions....