Forking and invariant measures in NIP theories

TL;DR

This paper presents an example in NIP theories where a formula does not fork over the empty set but has zero measure under all invariant measures, highlighting a distinction in measure behavior.

Contribution

It introduces a specific NIP theory example demonstrating that non-forking does not imply positive measure under invariant measures, contrasting with first-order amenable theories.

Findings

Existence of formulas with non-forking but zero measure

Difference in measure behavior between NIP and amenable theories

Illustration of measure properties in model theory

Abstract

We give an example of an NIP theory $T$ in which there is a formula that does not fork over $\varnothing$ but has measure $0$ under any global $\varnothing$-invariant Keisler measure, and we show that this cannot occur if $T$ is also first-order amenable.