A nonsingular action of the full symmetric group admits an equivalent invariant measure

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