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

Paolo Marimon

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arXiv · 2208.06323

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

Paolo Marimon

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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 The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.

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BibTeX RIS

Paolo Marimon. 2022-08-12. On the non-measurability of $\omega$-categorical Hrushovski constructions. https://arxiv.org/abs/2208.06323

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