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

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