  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.
