Keisler Measures and Generically Stable Random Types

Abstract

We introduce the notions of $rgs$ and $irgs$ for Keisler measures, motivated by the study of generically stable random types and their associated Morley sequences. We obtain characterizations of these notions in terms of averages of classical first-order formulas over suitable probabilistic partitions (Theorems 3.2 and 3.3). We compare these notions with $fim$, $fam$, and self-averaging, and show that for types the notions $fim$, $irgs$, and $rgs$ coincide. We prove that every $irgs$ measure is dependent (Theorem 4.5); consequently, such measures are symmetric (Corollary 4.8). Furthermore, we show that for $irgs$ measures the model-theoretic instability events ${\mathbf{O}}^{\ddot{I}}†$, ${\mathbf{I}}^{\ddot{I}}†$, and ${\mathbf{L}}^{\ddot{I}}†$ have ${\mathbb{P}}_{\hat{I}}¼$-measure zero (Theorem 5.4), extending results from [8] beyond the $fim$ case.

28 pages. This version contains minor revisions, including additional explanations in several places for greater clarity. Comments welcome. k.khanaki @ gmail.com