Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition. We propose various definitions, and show that all of them are equivalent and that for doubling spaces these are also equivalent to the well-known measured Gromov-Hausdorff convergence. Then we show that the curvature conditions CD(K, infinity) and RCD(K, infinity) (Riemannian curvature dimension, RCD) are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the L-2-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted Gamma-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the RCD(K, infinity) condition with K > 0. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.