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

Abstract

We prove that, given an $RC{D}^{\ast }(K,N)$-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 $m└R_i$ 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.

Final version to appear in the Annali della Scuola Normale Superiore Classe di Scienze

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