Gromov-Hausdorff and intrinsic flat convergence of RCD(K,N) and Kato spaces

We consider metric measure spaces $(X,\mathsf{d},{\mathcal{H}}^N)$ satisfying the properties (ETR), (LBD), and with an almost everywhere connected regular set. In particular, these assumptions are fulfilled by non-collapsed RCD$(K,N)$ spaces without boundary, as well as by non-collapsed strong Kato limit spaces without boundary. For both classes, we study orientability in the sense of metric currents, establish stability of orientation under pointed Gromov–Hausdorff convergence, and show that the pointed Gromov–Hausdorff limit coincides with the local flat limit.