Let $\Omega$ be one of $\X^{N+1},C[0,1],D[0,1]$: product of Polish spaces, space of continuous functions from $[0,1]$ to $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$, respectively. 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 $\Gamma(x) \subset \kP(\Omega)$, where $\kP(\Omega)$ 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. We show that such probability measure exists if and only if the $\alpha$ average of the so-called $G$-expectation of bounded continuous functions with respect to the measures in $\Gamma$ is less than their $\beta$ average.
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