The first part of the article mainly gives some concepts and lemmas, such as the definition of some function spaces, Radon integral and its basic properties, $L^1-$seminorm, the definition of Lebesgue integral and so on. These basic concepts and lemmas are very helpful in proving the main theorem later. The proof of the main theorem is divided into two parts. The main idea is to locally approximate the transformation $\phi:U\to V$ around a point $a\in U$ by an affine map. In order to be able to use this local approximation meaningfully, we decompose the given function $\psi\in C_c\left(V\right)$ into a sum of functions with very small support, so that the approximation of $\phi:U\to V$ by the local affine approximation is very good. This is done with the help of the $\zeta-$function introduced below, which provides a practical partition of the one on $\mathbb{R}^n$. In the second part, the proof will be performed on suitable approximations of any integrable function.