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Keisler measures in the wild

Gabriel Conant, Kyle Gannon and James Hanson

Vol. 2 (2023), No. 1, 1–67
DOI: 10.2140/mt.2023.2.1

We investigate Keisler measures in arbitrary theories. Our initial focus is on Borel definability. We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and satisfy associativity. However, we also demonstrate failures of both properties over uncountable parameter sets. In particular, we show that the Morley product of Borel definable types need not be Borel definable (correcting an erroneous result from the literature). We then study various notions of generic stability for Keisler measures and generalize several results from the NIP setting to arbitrary theories. We also prove some positive results for the class of frequency interpretation measures in arbitrary theories, namely, that such measures are closed under convex combinations and commute with all Borel definable measures. Finally, we construct the first example of a complete type which is definable and finitely satisfiable in a small model, but not finitely approximated over any small model.

Keisler measures, generic stability, neostability
Mathematical Subject Classification
Primary: 03C45, 03C95
Received: 24 March 2022
Revised: 26 November 2022
Accepted: 30 January 2023
Published: 11 June 2023
Gabriel Conant
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom
The Ohio State University
Columbus, OH
United States
Kyle Gannon
Department of Mathematics
University of California
Los Angeles, CA
United States
James Hanson
Department of Mathematics
University of Wisconsin
Madison, WI
Department of Mathematics
University of Maryland
College Park, MD
United States