Let $(M_j,g_j)$ be closed $n$-dimensional manifolds with $\operatorname{Ric}_{g_j}\ge (n-1)K g_j$, $\operatorname{diam}(M_j,g_j)\le D$, and $\operatorname{Vol}_{g_j}(M_j)\ge v>0$. Suppose $M_j\to X$ in Gromov-Hausdorff distance and choose Borel $\varepsilon_j$-approximations $\Phi_j:M_j\to X$ with $\varepsilon_j\to 0$, as in the standard measured Gromov-Hausdorff formulation. After passing to a subsequence, the pushforward measures $\mu_j=(\Phi_j)_*\operatorname{Vol}_{g_j}$ converge weakly to a Radon measure $\mu$ on $X$. The measure $\mu$ satisfies the Bishop-Gromov volume monotonicity inequality with curvature parameter $K$: for every $x\in X$ and every $0<r<R$, with $R<\pi/\sqrt K$ when $K>0$, one has

\begin{align*} \frac{\mu(B_X(x,R))}{V_{K,n}(R)} \leq \frac{\mu(B_X(x,r))}{V_{K,n}(r)}, \end{align*}

where $V_{K,n}(\rho)$ denotes the volume of a radius-$\rho$ ball in the simply connected $n$-dimensional space form of sectional curvature $K$. In the noncollapsed Ricci-limit setting there is a constant $c>0$ such that $\mu=c\,\mathcal H^n$ on $X$.