  We obtain some classification result for the mapping class groups of compact
orientable surfaces in terms of measure equivalence. In particular, the mapping
class groups of different closed surfaces can not be measure equivalent.
Moreover, we give various examples of discrete groups which are not measure
equivalent to the mapping class groups. In the course of the proof, we
investigate amenability in a measurable sense for the actions of the mapping
class group on the boundary at infinity of the curve complex and on the
Thurston boundary. Using this investigation, we prove that the mapping class
group of a compact orientable surface is exact.
