On definable amenability of definable groups in simple theories (Model theoretic aspects of the notion of independence and dimension)

Abstract

By Stone's representation theorem, We can identify sets of all types (over some set) with Stone spaces having a base of sets of all formulas over the set. So we can naturally extend the topologies to Borel regular probability measures. The measure is called Keisler measure. Keisler measures have been mainly studied under an assumption of NIP. In NIP, we can consider a notion of Vpnik-Chervonenkis dimension. Therefore, we can approximate Keisler measures as distributions of finitely many types by Vapnik-Chervonenkis theorem. In this paper, we introduce Keisler measures under an assumption of simple from NIP.