Sequential approximations for types and Keisler measures

TL;DR

This paper introduces the concepts of sequentially approximated types and Keisler measures, showing their relevance in various theories and providing new characterizations and generalizations in model theory.

Contribution

It defines sequential approximation notions for types and measures, and demonstrates their applicability to generically stable types and finitely satisfiable measures in NIP theories.

Findings

Generically stable types are sequentially approximated.

Keisler measures finitely satisfiable over a countable model are sequentially approximated.

Characterization of generically stable measures via smooth sequences.

Abstract

This paper is a modified chapter of the author's Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a sequence of realized types and measures which can be approximated by a sequence of `averaging measures' on tuples of realized types. We show that both generically stable types (in arbitrary theories) and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated. We also introduce the notion of a smooth sequence in a measure over a model and give an equivalent characterization of generically stable measures (in NIP theories) via this definition. In the last section, we take the opportunity to generalize the main result of [8].