Invariant measures on products and on the space of linear orders

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, $\mathrm{Aut}(M)$-invariant measure on $[0,1{]}^M$ is a product measure. We also investigate the action of $\mathrm{Aut}(M)$ on the compact space $\mathrm{LO}(M)$ of linear orders on $M$. If we assume moreover that the action $\mathrm{Aut}(M)\curvearrowright M$ is transitive, we prove that the action $\mathrm{Aut}(M)\curvearrowright \mathrm{LO}(M)$ either has a fixed point or is uniquely ergodic.