Let $g_\epsilon, f : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the familly $g_\epsilon$ is indexed by $\epsilon \in [0,1]$. $f$ is continuous on $K$ and $||g_\epsilon - g_0||_{L^\infty(K)} \le c \epsilon$ where $c$ does not depend on $\epsilon$, and $g_\epsilon$ are uniformly Lipschitz. We could also use that $||\nabla(g_\epsilon - g_0)||_{L^\infty(K)} \le c \epsilon$. How to prove that \begin{align*} \left| \int_{g_\epsilon^{-1}(\mu)} \nabla g_0 - \int_{g_0^{-1}(\mu)} \nabla g_0 \right| \le c \epsilon \end{align*} where $c$ does not depend on $\epsilon$ ?