  Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,\omega)$
the group of measure preserving homeomorphisms of $M$, where $\omega$ is a
finite measure induced by a volume form. In this paper we define volume and
Euler classes in bounded cohomology of an infinite dimensional transformation
group $\operatorname{Homeo}_0(M,\omega)$ and $\operatorname{Homeo}(M,\omega)$
respectively, and in several cases prove their non-triviality. More precisely,
we define:
  - Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,\omega))$
where $M$ is a hyperbolic manifold of dimension $n$.
  - Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,\omega))$
where $S$ is a closed hyperbolic surface.
  We show that Euler classes have positive norms for any closed hyperbolic $S$
and volume classes have positive norms for all hyperbolic surfaces and certain
hyperbolic $3$-manifolds, and hence they are non-trivial.
