Before revision:  We first 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 } (\Omega), where (\Omega) is the set of probability measures on \Omega.
After revision:  We first 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 }  (\Omega) is the set of probability measures on \Omega.