Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter

Abstract

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the CD(K,N) condition for N<0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance diKRW, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K,N) condition with N<0 is converging with respect to the distance diKRW to some metric measure space, then this limit structure is still a CD(K,N) space.