Non-collapsed spaces with Ricci curvature bounded from below

Abstract

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for $\mathsf{RCD}$ spaces. In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence.

Clarified the proof of Proposition 3.2, in particular step 2.2 (thank to Liming Yin for pointing this out)