A Lorentzian analog for Hausdorff dimension and measure

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the setting, this allows us to geometric dimension - akin Hausdorff for metric spaces that distinguishes between e.g. spacelike and null subspaces Minkowski spacetime. The measure corresponding its gives natural reference synthetic or limiting spacetime, what it means such spacetime be collapsed (in analogy with geometry theory Riemannian Ricci limit spaces). As crucial tool we introduce doubling condition causal diamonds notion measures. Moreover, applications continuous spacetimes connections timelike curvature bounds are given.