  Based on the multidimensional irreducible paving of De March & Touzi, we
provide a multi-dimensional version of the quasi sure duality for the
martingale optimal transport problem, thus extending the result of Beiglb\"ock,
Nutz & Touzi. Similar, we also prove a disintegration result which states a
natural decomposition of the martingale optimal transport problem on the
irreducible components, with pointwise duality verified on each component. As
another contribution, we extend the martingale monotonicity principle to the
present multi-dimensional setting. Our results hold in dimensions 1, 2, and 3
provided that the target measure is dominated by the Lebesgue measure. More
generally, our results hold in any dimension under an assumption which is
implied by the Continuum Hypothesis. Finally, in contrast with the
one-dimensional setting of Beiglb\"ock, Lim & Obloj, we provide an example
which illustrates that the smoothness of the coupling function does not imply
that pointwise duality holds for compactly supported measures.
