  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.
