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 $\mathbf{O}^\varphi$, $\mathbf{I}^\varphi$, and $\mathbf{L}^\varphi$ have $\mathbb{P}_\mu$-measure zero (Theorem 5.4), extending results from [8] beyond the $fim$ case.