Generic stability, randomizations, and NIP formulas

TL;DR

This paper explores the relationships between Keisler measures, generic stability, randomizations, and NIP formulas, introducing new concepts and characterizations to deepen understanding of model-theoretic stability properties.

Contribution

It introduces the notion of Keisler-Morley measures, proves properties of measures under randomization, and characterizes NIP formulas through tame extensions and stability conditions.

Findings

Keisler-Morley measures serve as Morley sequences for Keisler measures.

The map from definable measures to types in the randomization commutes with Morley products.

NIP formulas are characterized by the existence of tame global extensions for local measures.

Abstract

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if $\mu$ is fim over $M$, then for any Keisler-Morley measure $\lambda$ in $\mu$ over $M$ and any formula $\varphi(x,b)$, $\lim_{i \to \infty} \lambda(\varphi(x_i,b)) = \mu(\varphi(x,b))$. We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure $\mu$ to a type $r_\mu$ in the randomization. We prove that this map commutes with Morley products, and that if $\mu$ is fim then $r_\mu$ is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP$_2$ theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures.