*Accepted Paper*

**Inserted:** 1 sep 2016

**Last Updated:** 17 oct 2016

**Journal:** Commun. Contemp. Math.

**Year:** 2016

**Abstract:**

In this note we prove that in a metric measure space $(X,d,m)$ verifying the measure contraction property with parameters $K \in \mathbb{R}$ and $1< N< \infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $m$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

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