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 < ∞. For the first marginal measure, we assume that μ0 ≪ 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 mmeasure 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.