Non-singular and probability measure-preserving actions of infinite permutation groups
Todor Tsankov
TL;DR
The paper advances the ergodic theory of infinite permutation groups by showing that non-singular actions of non-archimedean Roelcke precompact groups on σ-finite spaces reduce to measure-preserving actions on open subgroups, via a decomposition into induced actions. It also proves a De Finetti–type rigidity result for invariant ergodic measures on shift spaces under a primitive, no-algebraicity setting with a necessary uniformity condition, offering a probabilistic path to independence that complements representation-theoretic methods. The approach blends unitary representation theory, model-theoretic algebraic closure, and tail-field analysis to yield structural decompositions, exact classifications, and product-measure conclusions with broad applicability to invariant random subgroups and exchangeability in 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 $(Ω, μ)$ admits an invariant $σ$-finite measure equivalent to $μ$. 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.