*Accepted Paper*

**Inserted:** 8 jul 2016

**Last Updated:** 20 jan 2017

**Journal:** Annali SNS

**Year:** 2016

**Doi:** 10.2422/2036-2145.201608_007

**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.

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