Let $D$ be a bounded Lipschitz domain and $f$ is continuous up to $\partial D$. Is it true that $$\int_{\partial D}f(x)d\sigma(x) = \lim_{\epsilon\to 0}\frac{1}{\epsilon}\int_{D^{\epsilon}}f(x)dx$$ where $D^{\epsilon}=\{y\in D: d(y,\partial D)<\epsilon\}$?

When $\partial D$ is $C^2$, we can parametrize each point of $D^{\epsilon}$ by a point $(\xi,\delta)\in \partial D\times [0,\epsilon)$ bijectively, when $\epsilon$ is small enough. But for Lipschitz it is not clear to me. Thanks!