Remarks on generic stability and random types

TL;DR

This paper introduces new properties of Keisler measures, $rgs$ and $irgs$, linking them to generically stable types and comparing them with existing concepts like $fim$, $fam$, and self-averaging.

Contribution

It defines $rgs$ and $irgs$ for Keisler measures and establishes their equivalence to the existence of certain generically stable types, advancing understanding of measure stability.

Findings

$irgs$ measures are dependent.

$rgs$ and $irgs$ are equivalent to the existence of generically stable extensions.

Comparison with $fim$, $fam$, and self-averaging concepts.

Abstract

We introduce the notions of $rgs$ and $irgs$ as properties of a Keisler measure $\mu$, and prove that they are respectively equivalent to the existence of a generically stable random type that extends $\mu$ and to the fact that its canonical extension, namely the random type $r_\mu$, is generically stable. We compare these notions with the known concepts of $fim$, $fam$, and self-averaging, and in particular we show that every $irgs$ measure is dependent in the sense of [10].