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

Karim Khanaki¹

¹ Faculty of Fundamental Sciences Arak University of Technology 38135-1177, Arak, Iran and School of Mathematics Institute for Research in Fundamental Sciences 19395-5746, Tehran, Iran

Fundamenta Mathematicae, Tome 234 (2016) no. 3, pp. 253-286

Résumé

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.

Détail

DOI : 10.4064/fm208-1-2016

Keywords: paper has three parts first study characterize amenable extremely amenable topological semigroups terms invariant measures using integral logic prove definability properties topological semigroup amenability fixed point compacta property second define types develop local stability framework integral logic stable formula phi prove definability complete phi types models deduce fundamental theorem stability third study important property measure theory talagrand stability point out connection between talagrand stability dependence property nip prove measure theoretic version definability types nip formulas