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Generically stable Keisler measures

  • Gabriel Conant
  • Kyle Gannon
  • James E. Hanson
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Abstract

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" rμ is generically stable in the randomization theory TR; (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)⇒(ii)⇒ (iii) were previously established by the authors (for T discrete). The primary focus of this paper is the reverse implications (iii)⇒ (ii)⇒(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.

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Cite

BibTeXRIS
BibTeX
@misc{conant2026generically,
  title = {{Generically stable Keisler measures}},
  author = {Gabriel Conant and Kyle Gannon and James E. Hanson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.24605},
  eprint = {2608.24605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.24605},
}
RIS
TY  - UNPB
TI  - Generically stable Keisler measures
AU  - Gabriel Conant
AU  - Kyle Gannon
AU  - James E. Hanson
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.24605
UR  - https://arxiv.org/abs/2608.24605
AB  - 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)⇒(ii)⇒ (iii) were previously established by the authors (for T discrete). The primary focus of this paper is the reverse implications (iii)⇒ (ii)⇒(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.
ER  -