Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 1, pp. 46-54   https://doi.org/10.15407/mag16.01.046     ( to contents , go back )

### 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 $\mu$.

Mathematics Subject Classification 2000: 37A40, 22A25, 22F10.
Key words: full symmetric group, nonsingular automorphism, Koopman representation, invariant measure.