There is a constant $\upsilon>0$ such that

$$\nu(X)=\upsilon\int_{M}\omega_{M}^{n},\ \ \ \ \underline{V}_{0}(x,r)=\upsilon\int_{f^{-1}(B_{\omega}(\phi^{-1}(x),r))}\omega_{M}^{n},$$

whenever $x\in X_{0}$ and $r\leqslant 1$ is such that $B_{\omega}(\phi^{-1}(x),2r)$ is a geodesically convex subset of $(N_{0},\omega)$.