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

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.