Biinvariant functions on the group of transformations leaving a measure quasiinvariant

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.