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arXiv · 2109.11760

MS-measurability via Coordinatization

Mostafa Mirabi
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Abstract

We prove that Macpherson-Steinhorn measurability is preserved under finite nil-interaction tree products. If the canonical component theories are MS-measurable and their universes are normalized to have dimension one, then every admissible nil-interaction coordinatized structure is MS-measurable. The resulting dimension is the sum of the local component dimensions appearing in the tree closure, and the measure is the corresponding product of local measures. We also prove a structural converse: in the countable $\aleph_0$-categorical setting, admissible nil-interaction structures are precisely definable expansions of their canonical tree products. Thus nil-interaction exactly isolates the tree-product case of coordinatization. Further preservation results show that canonical tree products preserve supersimplicity of finite SU-rank and one-basedness, with SU-rank computed by the same local sum formula. Finite homogeneous component approximations also combine into finite tree envelopes, so smooth approximability is preserved. Finally, enriched finite tree products of multidimensional asymptotic, respectively exact, component classes are again multidimensional asymptotic, respectively exact, up to the natural weak/reduct distinction.

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Mostafa Mirabi. 2021-09-24. MS-measurability via Coordinatization. https://arxiv.org/abs/2109.11760

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