Let $(M_{j}^{m},g_{j},p_{j})$ be a sequence of Riemannian manifolds with $\Ric_{g_{j}}\geq-(m-1)$ such that

(6.136)

$$(M_{j}^{m},g_{j},p_{j})\xrightarrow{GH}(X_{\infty},d_{\infty},p_{\infty})$$

for some metric space $(X_{\infty},d_{\infty},p_{\infty})$, then by passing to a subsequence, the renormalized measures

(6.137)

$$d\underline{\nu}_{j}\equiv\frac{\dvol_{g_{j}}}{\Vol_{g_{j}}(B_{1}(p_{j}))}$$

converge to a Radon measure $d\underline{\nu}_{\infty}$ on $X_{\infty}$ which is called the renormalized limit measure. The Gromov-Hausdorff convergence together with the convergence of the renormalzied measures is called the measured Gromov-Hausdorff convergence.