Before revision:  We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
After revision:  We consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset\kP\kP .