Biinvariant functions on the group of transformations leaving a measure quasiinvariant

TL;DR

This paper characterizes the structure of certain transformation groups on measure spaces and shows that all continuous biinvariant functions are determined by Radon--Nikodym derivatives.

Contribution

It provides a canonical form for double cosets in the transformation group and links biinvariant functions to Radon--Nikodym derivatives.

Findings

Canonical forms of double cosets are described.

Continuous biinvariant functions depend on Radon--Nikodym derivatives.

The structure of the transformation group is clarified.

Abstract

Let $Gms$ be the group of transformations of a Lebesgue space leaving the measure quasiinvariant, let $Ams$ be its subgroup consisting of transformations preserving the measure. We describe canonical forms of double cosets of $Gms$ by the subgroup $Ams$ and show that all continuous $Ams$-biinvariant functions on $Gms$ are functionals on of the distribution of a Radon--Nikodym derivative.