The talk presents some recent advances in the theory of Lorentzian length spaces, starting from the articles [1] and [2] and the complementary approach [3]. After giving an account of the results of those articles, we first discuss stability of causal and curvature properties under Lorentzian Gromov-Hausdorff convergence. Then we present the recently obtained (positive) answer to the question raised in [2] of the denseness of the subspace of all connected n-dimensional causal spacetimes in the space of all connected Lorentzian length spaces with local Hausdorff dimension bounded above by n. This result implies that the functors from [1] and [4] between ordered measure spaces and Lorentzian length spaces are indeed inverses to each other on the space of all Lorentzian length spaces with bounded local Hausdorff dimension. If time permits, we also discuss natural geometric conditions under which the Lorentzian Gromov-Hausdorff measure is Radon, a relevant step in making Lorentzian length spaces accessible to differential calculus, in analogy to metric (measure) spaces.
[1] O. Müller: Functors in Lorentzian geometry - three variations on a theme, arXiv:2205.01617
[2] O. Müller: Gromov-Hausdorff distances for Lorentzian length spaces, arXiv:2209.12736
[3] E. Minguzzi, S. Suhr: Lorentzian metric spaces and their Gromov-Hausdorff convergence, arXiv:2209.14384
[4] R. McCann, C. Sämann: Gromov-Hausdorff measure for Lorentzian length spaces, arXiv:2110.04386