Dependent measures in independent theories

Abstract

We introduce the notion of dependence, as a property of a Keisler measure, and generalize some results of Hrushovski, Pillay & Simon (2013) in NIP theories (theories satisfying the negation of the independence property) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons:

all measures in NIP theories are dependent, all types and all frequency interpretation measures (fims) in any theory are dependent, and as a crucial result in measure theory, the Glivenko–Cantelli class of functions (formulas) is characterized by dependent measures.