Invariant measures on products and on the space of linear orders

Colin Jahel; Todor Tsankov

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arXiv · 2007.00281

Invariant measures on products and on the space of linear orders

Colin Jahel·Todor Tsankov

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Abstract

Let $M$ be an $\aleph_0$-categorical structure and assume that $M$ has no algebraicity and has weak elimination of imaginaries. Generalizing classical theorems of de Finetti and Ryll-Nardzewski, we show that any ergodic, $\operatorname{Aut}(M)$-invariant measure on $[0, 1]^M$ is a product measure. We also investigate the action of $\operatorname{Aut}(M)$ on the compact space $\mathrm{LO}(M)$ of linear orders on $M$. If we assume moreover that the action $\operatorname{Aut}(M) \curvearrowright M$ is transitive, we prove that the action $\operatorname{Aut}(M) \curvearrowright \mathrm{LO}(M)$ either has a fixed point or is uniquely ergodic.

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BibTeX RIS

Colin Jahel, Todor Tsankov. 2020-07-01. Invariant measures on products and on the space of linear orders. https://arxiv.org/abs/2007.00281

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