An invitation to extension domination

TL;DR

This paper introduces a new concept called extension domination for Keisler measures, extending the existing theory of domination for types and establishing foundational properties and examples.

Contribution

It defines extension domination for Keisler measures, proves its key properties, and connects it to existing domination concepts for types.

Findings

Extension domination extends type domination.

It forms a preorder on global Keisler measures.

Basic properties like approximation and closure are established.

Abstract

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 type setting 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.