Invariant Keisler Measures for ω-Categorical Structures

Abstract

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay and Ramsey finds the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. In this article we give the first known simple ω-categorical counterexamples. These happen to be various ω-categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple ω-categorical structures where a formula forks over ∅ if and only if it is universally measure zero must satisfy a stronger version of the independence theorem.