Kyle Gannon

Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems and provide some explicit examples.

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.