Monge problem in metric measure spaces with Riemannian curvature-dimension condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X, d, m) enjoying the Riemannian curvature-dimension condition RCD* (K, N), with N < infinity. For the first marginal measure, we assume that mu(0) << m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.