Year: 2026
Type: preprint
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 if a definable global Keisler measure$μ$ is not dependent, then every Morley sequence in its canonical random extension $r_μ$ is divergent (Theorem 4.7). Consequently, every $irgs$ measure is dependent (Corollary 4.8) and hence symmetric (Corollary 4.13). Furthermore, we show that for $irgs$ measures the model-theoretic instability events $\mathbf{O}^φ$, $\mathbf{I}^φ$, and $\mathbf{L}^φ$ have $\mathbb{P}_μ$-measure zero (Theorem 5.4). In view of the equivalence of $fim$ and $irgs$ subsequently established in [6], this provides an alternative proof of the corresponding result for $fim$ measures in [9].
Source: arXiv (Cornell University)
Author: Khanaki, Karim
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3058533074
OpenAlex ID: W7161590571