Generic stability, randomizations, and NIP formulas

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.