Optimal maps in essentially non-branching spaces

In this note we prove that in a metric measure space (X,d,m) verifying the measure contraction property with parameters K∈R and 1<∞, 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.