  We define a notion of a measured length space X having nonnegative N-Ricci
curvature, for N finite, or having infinity-Ricci curvature bounded below by K,
for K a real number. The definitions are in terms of the displacement convexity
of certain functions on the associated Wasserstein 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.
