Kyle Gannon

In an important (yet unpublished) research note, Ben Yaacov describes how to turn global Keisler measures into types over a monster model of the randomization. These transfer methods allow one to turn questions involving measures into those involving types (in continuous logic). Assuming that T is NIP, we show that the Morley product commutes with the transfer map for finitely satisfiable measures. We characterize when the Morley product commutes with the restriction map for pairs of global finitely satisfiable types in the randomization. We end by making some brief observations about the Ellis semigroup in this context.

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler–Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if [Formula: see text] is fim over [Formula: see text], then for any Keisler–Morley measure [Formula: see text] in [Formula: see text] over [Formula: see text] and any formula [Formula: see text], [Formula: see text]. We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure [Formula: see text] to a type [Formula: see text] in the randomization. We prove that this map commutes with Morley products, and that if [Formula: see text] is fim then [Formula: see text] is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP2 theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures.

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.