Non-singular and Probability Measure-Preserving Actions of Infinite Permutation Groups

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 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 ZM, where Z is a Polish space, is a product measure.