Ricci curvature for metric-measure spaces via optimal transport

We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N is an element of [1, infinity), or having infinity-Ricci curvature bounded below by K, for K is an element of R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P-2 (X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.