  By disintegration of transport plans it is introduced the notion of transport
class. This allows to consider the Monge problem as a particular case of the
Kantorovich transport problem, once a transport class is fixed. The transport
problem constrained to a fixed transport class is equivalent to an abstract
Monge problem over a Wasserstein space of probability measures. Concerning
solvability of this kind of constrained problems, it turns out that in some
sense the Monge problem corresponds to a lucky case.
