We study the space of normalized metric measure spaces (M,d,m) and 
introduce a complete separable metric D on it. This metric has a natural 
interpretation in terms of mass transportation.
It turns out that the family of normalized metric measure spaces with 
doubling constant \le C is closed under D-convergence. Moreover, the 
subfamily of spaces with diameter \le R is compact.

Furthermore, we introduce and analyze curvature bounds for metric 
measure spaces (M,d,m), based on convexity properties of the relative 
entropy Ent(.|m). For Riemannian manifolds, Curv(M,d,m) \ge  K if and 
only if  Ric_M(v,v) \ge K |v|^2 for all v \in TM.
Our lower curvature bounds are stable under D-convergence.