A Nonsingular Action of the Full Symmetric
Group Admits an Equivalent Invariant
Measure

Nikolay Nessonov

B.Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: nessonov@ilt.kharkov.ua

Received November 11, 2018, revised October 9, 2019.

Abstract

Let $\overline{\mathfrak{S}}_\infty$ denote the set of all
bijections of natural numbers. Consider an action of
$\overline{\mathfrak{S}}_\infty$ on a \emph{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
$\mu$.