On the non-measurability of \(\omega \)-categorical Hrushovski constructions

We study \(\omega \)-categorical MS-measurable structures. Our main result is that a certain 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 constrast 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 The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.