Non-singular and probability measure-preserving actions of infinite permutation groups

Abstract

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega, \mu)$ admits an invariant $\sigma$-finite measure equivalent to $\mu$. Second, we prove the following de Finetti type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^M$, where $Z$ is a Polish space, is a product measure.