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 $\mu$.