Invariant Keisler measures for omega-categorical structures

TL;DR

This paper presents the first known simple omega-categorical structures that serve as counterexamples to the notion that formulas not forking over the empty set are measure zero, expanding understanding of measure and independence in model theory.

Contribution

It introduces the first simple omega-categorical counterexamples using Hrushovski constructions and explores their measure-theoretic properties.

Findings

Counterexamples of simple omega-categorical structures with non-forking formulas that are measure zero.

A probabilistic independence theorem linking forking and measure zero ideals.

Stronger independence properties in structures where forking and measure zero ideals coincide.

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 $\emptyset$ but are universally measure zero. In this article we give the first known simple $\omega$-categorical counterexamples. These happen to be various $\omega$-categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $\omega$-categorical structures where the forking ideal and the universally measure zero ideal coincide must satisfy a stronger version of the independence theorem.