Biinvariant functions on the group of transformations leaving a measure quasiinvariant

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.