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

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. (c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).