On the bounded cohomology for ergodic nonsingular actions of amenable groups

Let Gamma be an amenable countable discrete group. Fix an ergodic free non-singular action of Gamma on a nonatomic standard probability space. Let G be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of G is compact in the natural topology. It is shown that there exists a bounded ergodic G-valued cocycle of Gamma.