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

TL;DR

This paper establishes new results in ergodic theory for infinite permutation groups, showing invariant measures exist for certain actions and characterizing invariant measures as product measures under specific conditions.

Contribution

It generalizes existing theorems to broader classes of groups and actions, providing new insights into measure-preserving properties and invariant measures.

Findings

Every non-singular action of certain Polish groups admits an equivalent invariant measure.

Invariant measures for primitive permutation groups under specified conditions are product measures.

The results extend ergodic theory to non-archimedean, Roelcke precompact 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.