标题： Hausdorff 维数和测度的洛伦兹类比 显示英文标题

标题： A Lorentzian analog for Hausdorff dimension and measure

摘要： 我们定义了一类关于洛伦兹(pre-)度量空间的单参数族规范体积测度。 在洛伦兹背景下，这使我们可以定义一个几何维度——类似于度量空间中的豪斯多夫维数——从而区分闵可夫斯基时空中的类空子空间和零子空间等。 与该几何维度对应的体积测度给出了合成或极限时空上的自然参考测度，并允许我们定义此类时空塌缩的意义（类比于度量测度几何和黎曼Ricci极限空间理论）。 作为关键工具，我们引入了因果钻石的加倍条件以及因果加倍测度的概念。 此外，还讨论了连续时空的应用以及与合成时像曲率界限的关系。 显示英文摘要

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