On the volume measure of non-smooth spaces with Ricci curvature bounded below

Abstract: We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i}){i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.