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C. Griessler:
"An extended footnote on finitely minimal martingale measures";
arXiv.org, June 9 (submitted) (2016), 7 S.

This note contains a short discussion on the sufficiency of
finite optimality in martingale transport. It is shown that fi
nitely minimal martingale measures are solutions of the martingale tra
nsport problem when the cost function is upper semi-continuous and bounded from above by a sum of integrable functions. As an application a transparent proof of the uniqueness of left-monotone martingale transport plans is given.

martingales optimal transport monotonicity principle

https://arxiv.org/pdf/1606.03106.pdf