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. The main application we have in mind is the martingale optimal transport problem when the martingales are assumed to have bounded volatility/ quadratic variation.
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), where This paper focuses on martingale optimal transport problem when the martingales are assumed to have bounded volatility/ quadratic variation.