This article is cited in 1 paper

Bi-invariant functions on the group of transformations leaving a measure quasi-invariant

Yu. A. Neretin<sup>abc

a</sup> Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^c University of Vienna

Abstract: Let $\mathrm{Gms}$ be the group of transformations of a Lebesgue space leaving the measure quasi-invariant. Let $\mathrm{Ams}$ be a subgroup of it consisting of transformations preserving the measure. We describe canonical forms of double cosets of $\mathrm{Gms}$ by the subgroup $\mathrm{Ams}$ and show that all continuous $\mathrm{Ams}$-bi-invariant functions on $\mathrm{Gms}$ are functionals of the distribution of a Radon-Nikodym derivative.
Bibliography: 14 titles.

Keywords: Lebesgue space, transformations of measure spaces, Polish group, double cosets.

UDC: 517.986.6+517.987.1+512.546

MSC: Primary 22E66, 28D99, 22F10; Secondary 28E99

Received: 04.02.2014 and 08.06.2014

DOI: 10.4213/sm8339

 English version: Sbornik: Mathematics, 2014, 205:9, 1357–1372

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