On the non-measurability of $\omega$-categorical Hrushovski constructions

TL;DR

This paper demonstrates that certain $mbda$-categorical Hrushovski constructions with finite $SU$-rank are not $MS$-measurable, revealing limitations in their measure-theoretic properties and exploring related structural features.

Contribution

It proves non-measurability of a class of $mbda$-categorical Hrushovski structures with finite $SU$-rank and discusses their independence and dimension properties.

Findings

Certain $mbda$-categorical Hrushovski structures are not $MS$-measurable.

In these structures, $MS$-dimension aligns with $SU$-rank.

Non-forking independence implies probabilistic independence in the measure.

Abstract

We study $\omega$-categorical $MS$-measurable structures. Our main result is that a class of $\omega$-categorical Hrushovski constructions, supersimple of finite $SU$-rank is not $MS$-measurable. These results complement the work of Evans on a conjecture of Macpherson and Elwes. In contrast to Evans' work, our structures may satisfy independent $n$-amalgamation for all $n$. We also prove some general results in the context of $\omega$-categorical $MS$-measurable structures. Firstly, in these structures, the dimension in the $MS$-dimension-measure can be chosen to be $SU$-rank. Secondly, non-forking independence implies a form of probabilistic independence in the measure. The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.