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.
Categories: math.MG math.DG math.FA