Amenability, extreme amenability, model-theoretic stability, and dependence property in integral logic

TL;DR

This paper explores the application of integral logic to characterize amenability, stability, and dependence properties in topological semigroups and model theory, providing new definability results and connections between measure theory and logic.

Contribution

It introduces a framework using integral logic to characterize amenability, develop local stability, and connect Talagrand's stability with NIP, offering new definability results.

Findings

Characterization of amenable and extremely amenable semigroups via invariant measures.

Definability of types and fundamental stability theorem in integral logic.

Connection between Talagrand's stability and NIP, with measure-theoretic definability of types.

Abstract

This paper has three parts. First, we study and characterize amenable and extremely amenable topological semigroups in terms of invariant measures using integral logic. We prove definability of some properties of a topological semigroup such as amenability and the fixed point on compacta property. Second, we define types and develop local stability in the framework of integral logic. For a stable formula $\phi$, we prove definability of all complete $\phi$-types over models and deduce from this the fundamental theorem of stability. Third, we study an important property in measure theory, Talagrand's stability. We point out the connection between Talagrand's stability and dependence property (NIP), and prove a measure theoretic version of definability of types for NIP formulas.