Local Keisler Measures and Nip Formulas

Abstract

We study generically stable measures in the local, NIP context. We show that in this setting, a measure is generically stable if and only if it admits a natural finite approximation.

Remarks on generic stability in independent theories

“…In [15], the second author proved a local version of the equivalence of (i) and (ii) in Theorem 2.8. Specifically, if φ(x; y) is NIP and µ is a local Keisler measure on φ-formulas, which is dfs (suitably defined), then µ is finitely approximated.…”

“…The next fact lists some implications between the notions above. These are standard exercises (see also [29,Chapter 7] and [15,Proposition 4.12]).…”

“…Each of these notions extends naturally to the space of all local Keisler measures, analogous to Definition 2.1. See [15] for details.…”

“…However, if we shift our focus to the local level of φ-types and φ-measures, then interesting examples emerge. This viewpoint is also motivated by a result of the second author [15] that if φ(x; y) is an NIP formula, then any definable and finitely satisfiable Keisler measure on φ-definable sets is finitely approximated. In Section 5.3, we show that this fails for φ-types in simple theories.…”

Sequential approximations for types and Keisler measures

“…In the last section, we consider connections with the local measure case and generalize the main result in [8] (Theorem 6.4). Explicitly, the main result in [8] demonstrates that if a formula φ is NIP and µ is a φ-measure which is φ-definable and finitely satisfiable over a countable model, then µ is φ-finitely approximated in the said model. Here, we demonstrate that countable can be replaced by small.…”