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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 an oriented closed hyperbolic surface.
We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic 
$3$-manifolds; hence, they are non-trivial.
