发表机构
University of Illinois Chicago; Peking University; Iowa State University（伊利诺伊大学芝加哥分校; 北京大学; 爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一阶理论T的凯斯勒测度，证明了频率解释测度、可定义且其随机扩张r_μ在随机化理论T^R中一般稳定、自平均这三个条件等价，解决了相关长期研究目标，通过AI模型得到了反向蕴含关系。
AI 中文摘要
给定一阶理论T（离散或连续逻辑中的）及T中一个伯雷可定义的全局凯斯勒测度μ，本文证明以下条件等价：(i) μ是频率解释测度；(ii) μ是可定义的，且其典型“随机扩张”r_μ在随机化理论T^R中是一般稳定的；(iii) μ是“自平均的”。该结果为凯斯勒测度建立了鲁棒的一般稳定性概念，解决了前人工作中长期存在的研究目标。此前作者已证明蕴含关系(i)⇒(ii)⇒(iii)（针对离散T），本文核心聚焦于反向蕴含关系(iii)⇒(ii)⇒(i)，借助AI模型得到了该结果。
英文摘要
Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $μ$ in $T$, we show that the following conditions are equivalent: $(i)$ $μ$ is a frequency interpretation measure; $(ii)$ $μ$ is definable and its canonical "random extension" $r_μ$ is generically stable in the randomization theory $T^R$; $(iii)$ $μ$ 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.
Comments23 pages; v2 answers the main question left open in v1
