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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.
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L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ \sigma -finite measure
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D. Bate, Structure of measures in Lipschitz differentiability spaces
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G. De Philippis, A. Marchese and F. Rindler, On a conjecture of Cheeger
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