Generically Stable Measures and Distal Regularity in Continuous Logic

TL;DR

This paper extends the theory of generically stable measures to NIP metric theories, establishing key properties and characterizations of distality, and introduces analytic regularity results in continuous logic.

Contribution

It generalizes generically stable measures to NIP metric theories and characterizes distality through measure properties, providing new analytic regularity results.

Findings

Generically stable measures in NIP metric theories retain key properties.

Distality characterized by smoothness of all generically stable measures.

Established analytic versions of distal regularity and Erdős-Hajnal property.

Abstract

We develop a theory of generically stable and smooth Keisler measures in NIP metric theories, generalizing the case of classical logic. Using smooth extensions, we verify that fundamental properties of (Borel)-definable measures and the Morley product hold in the NIP metric setting. With these results, we prove that as in discrete logic, generic stability can be defined equivalently through definability properties, statistical properties, or behavior under the Morley product. We also examine weakly orthogonal Keisler measures, characterizing weak orthogonality in terms of various analytic regularity properties. We then examine Keisler measures in distal metric theories, proving that as in discrete logic, distality is characterized by all generically stable measures being smooth, or by all pairs of generically stable measures being weakly orthogonal. We then use this, together with our results on weak orthogonality and a cutting lemma, to find analytic versions of distal regularity and the strong Erd\H{o}s-Hajnal property.