TITLE: Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature
QUESTION [5 upvotes]: Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov compactness theorem says that there exists a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space $X$.
Questions. (1) Is the Hausdorff dimension of $X$ necessarily integer?
(2) Is it at most $n$?
Remark. If one assumes a stronger condition that the sectional (rather than Ricci) curvature of $M_i$ is uniformly bounded below then the answers to both questions are positive as it is shown in the theory of Alexandrov spaces.

REPLY [3 votes]: I cannot say anything about question (1), but the answer to question (2) is yes:
By the work of Cheeger-Colding, we can assume that $(M_i)$ endowed with the normalized volume measure converges in the measured Gromov-Hausdorff sense. Thus, the limit is a $\mathrm{CD}(K,n)$-space in the sense of Lott-Sturm-Villani (where $K$ is the uniform lower bound on the Ricci curvatures of $M_i$). These spaces have Hausdorff dimension $\leq n$.
You should find these results (or references to the original sources) in K.-T. Sturm. On the geometry of metric measure spaces I. and II.