On the measure contraction property of metric measure spaces

We introduce a measure contraction property of metric measure spaces which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds. It is actually equivalent to the lower bound of the Ricci curvature in the Riemannian case. We will generalize the Bonnet–Myers theorem, and prove that this property is preserved under the measured Gromov–Hausdorff convergence.