Dependent measures in independent theories

Abstract

We introduce the notion of dependence, as a property of a Keisler measure, and generalize several results of [HPS13] on generically stable measures (in $NIP$ theories) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons: (i) all measures in $NIP$ theories are dependent, (ii) all types and all $fim$ measures in any theory are dependent, and (iii) as a crucial result in measure theory, the Glivenko-Cantelli class of functions (formulas) is characterized by dependent measures.