Generically stable Keisler measures

Abstract

Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\mu$ in $T$, we show that the following conditions are equivalent: $(i)$ $\mu$ is a frequency interpretation measure; $(ii)$ $\mu$ is definable and its canonical "random extension" $r_{\mu }$ is generically stable in the randomization theory $T^R$; $(iii)$ $\mu$ is "self-averaging". This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\Rightarrow (ii)\Rightarrow (iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\Rightarrow (ii)\Rightarrow (i)$. We also prove that generically stable measures are closed under Morley products, answering another well-known question that was open even in the case of types. These results are obtained through the use of AI models.

arXiv comment23 pages; v2 answers the main question left open in v1