Remarks on generic stability in independent theories

Abstract

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as φ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.Date: May 28, 2019.

Associativity of the Morley Product of Invariant Measures in Nip Theories

“…The obvious question at this point is what happens when T is not NIP. In fact, several issues similar to those identified in the previous discussion were also noted by the authors while considering Borel-definable measures in arbitrary theories during the preparation of [1]. Ultimately, these obstacles were avoided by narrowing the focus to definable measures, which sufficed for the main results (see [1, Remark 2.11]).…”

“…In fact, several issues similar to those identified in the previous discussion were also noted by the authors while considering Borel-definable measures in arbitrary theories during the preparation of [1]. Ultimately, these obstacles were avoided by narrowing the focus to definable measures, which sufficed for the main results (see [1, Remark 2.11]). Following an earlier draft of this note, we resumed the general investigation of Borel-definable measures in [2] (joint with Hanson).…”

“…Note that in the previous argument, we only needed a weaker version of Proposition 3.1 in which all three measures are smooth. Since smooth measures are definable (by Proposition 1.2(a), one could instead use associativity for definable measures, which has a short and fairly elementary proof (see [1, Proposition 2.6]).…”

Sequential approximations for types and Keisler measures

“…The general notion of a generically stable type was introduced by Pillay and Tanović in [16]. The definition of a generically stable type provided below was proved to be equivalent in [6] (see Proposition 3.2 there). We also provide the definition of a dfs type which will be important throughout this paper.…”

“…Let L be a countable sublanguage of L ′ containing all the formulas in ∆. The corresponding type p| L is generically stable over the model M where M = M ′ | L (see [6,Remark 3.3]). Hence we may apply Theorem 4.8.…”