Let $\HH^m$ denote the Hausdorff measure. Let $m\le n$ and consider a Lipschitz function $f\colon \R^m \to \R^n$ . If $A\subset \R^m$ is a Lebesgue measurable set, the equation $$ \int_A J_f(x) \,dx = \int_{\R^n} \HH^0(f^{-1}(\{y\})\cap A) \, d\HH^m y $$ holds, where $$ J_f(x) = \sqrt{\det(Df(x)\cdot Df(x)^*)} $$ is the Jacobian determinant of $f$ in the point $x$ and represent the $m$ -volume of the image of the unit cube under the linear map $Df(x)$ .

If $u\in L^1(\R^m)$ then one has $$ \int_{\R^m} u(x) J_f(x)\, dx = \int_{\R^n}\sum_{x\in f^{-1}(\{y\})} u(x)\, d\HH^m y. $$

Notice that this formula is a generalization of the change of variables in integrals on $\R^n$ .