 bijection $\phi: M \to N$ between $n$-manifolds the following   is true:  $\phi^{-0}$ is a local diffeomorphism a.e. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-0}(U)} \phi^*\omega$.   