  We introduce the notion of measurable bounded cohomology for measured
groupoids, extending continuous bounded cohomology of locally compact groups.
We show that the measurable bounded cohomology of the semidirect groupoid
associated to a measure class preserving action of a locally compact group $G$
on a regular space is isomorphic to the continuous bounded cohomology of $G$
with twisted coefficients. We also prove the invariance of measurable bounded
cohomology under similarity. As an application, we compare the bounded
cohomology of (weakly) orbit equivalent actions and of measure equivalent
groups. In this way we recover an isomorphism in bounded cohomology similar to
one proved by Monod and Shalom. For amenable groupoids, we show that the
measurable bounded cohomology vanishes. This generalizes previous results by
Monod, Anantharaman-Delaroche and Renault, and Blank.
