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!



Surface integral approximation
This is really a question about [Minkowski content.](http://en.wikipedia.org/wiki/Minkowski_content) If you look at the cited article (under properties), this seems to indicate that under your Lipschitz condition, the answer to your question is YES (since the full-dimensional Hausdorff measure is the Lebesgue measure).


