Local curvature-dimension condition implies measure-contraction property

We prove that for non-branching metric measure spaces the local curvature condition CDloc(K,N) implies the global version of MCP(K,N). The curvature condition CD(K,N) introduced by the second author and also studied by Lott & Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. It is worth pointing out that our result implies sharp Bishop-Gromov volume growth inequality and sharp Poincar\'e inequality.