Let $(M^{n}_{j},g_{j},p_{j})$ be a sequence of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$, ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and such that

$\displaystyle(M_{j}^{n},d_{j},p_{j})\to(X,d,p)\,.$

(5.7)

Then there exists a subset, $\mathcal{S}\subseteq X$, with $\dim\mathcal{S}\leq n-3$, such that for each $x\in X\setminus\mathcal{S}$, we have $r_{h}(x)>0$. In particular, $x\in X\setminus\mathcal{S}$ is a $C^{1,\alpha}$ Riemannian manifold.