Any nonsingular action of the full symmetric group is isomorphic to an action with invariant measure

Abstract

Let ${\overline{\mathfrak{S}}}_{\infty }$ denote the set of all bijections of natural numbers. Consider the action of ${\overline{\mathfrak{S}}}_{\infty }$ on a measure space $\left(X,\mathfrak{M},\mu \right)$, where $\mu$ is ${\overline{\mathfrak{S}}}_{\infty }$-quasi-invariant measure. We prove that there exists ${\overline{\mathfrak{S}}}_{\infty }$-invariant measure equivalent to $\mu$.

10 pages