Keisler measures in the wild

TL;DR

This paper explores properties of Keisler measures in various theories, focusing on Borel definability, stability, and new examples, revealing both consistencies and failures of expected properties.

Contribution

It provides new insights into Borel definability, stability, and the behavior of Keisler measures in arbitrary theories, correcting previous misconceptions and establishing new results.

Findings

Borel definable measures are closed under Morley products in countable theories

Failures of closure properties over uncountable parameter sets

Constructs the first example of a type definable and finitely satisfiable but not finitely approximated

Abstract

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.