Transformation of Lebesgue Measure and Integral by Lipschitz Mappings

Abstract

We first show that {\sc Lipschitz} mappings transform measurable sets into measurable sets. Then we prove the following theorem:\par {\it Let $E\subseteq \mathbb{R}^n$ be open, and let $\phi: E\to\mathbb{R}^n$ be continuous. If $\phi$ is differentiable at $x_0\in E$, then} \[ \lim\limits_{r\to 0} \dfrac{\lambda_n(\phi(B_r(x_0)))}{\lambda_n(B_r)} = \Big| \det\phi'(x_0)\Big|. \] From this result the change of variables formula for injective and locally {\sc Lipschitz} mappings is easily derived by using the {\sc Radon-Nikodym} theorem. We finally discuss the transformation of $L^p$ functions by {\sc Lipschitz} mappings.