  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.
