Let $D\subset R^d$ be a bounded Lipschitz domain. Must there exist a bounded function $\Phi$ on $\partial D$ and collections of subsets $(\partial D )^{\epsilon} \subset \partial D $ (indexed by $\epsilon$) such that for all bounded continuous function $g$ on $\partial D$ we have $\lim_{\epsilon\to 0} \sum_{(\partial D )^{\epsilon}}g\,\Phi\,\epsilon^{d-1}= \int_{\partial D }g\,d\sigma $
We may suppose $\partial D$ is just the graph of a Lipschitz function over a ball in $R^{d-1}$.
This is a discrete approximation to the surface measure $\sigma$. The result is true if we add the condition that "the unit outward normal vector field on $\partial D $ is continuous $\sigma-$almost everywhere on $\partial D $." But I wonder if we can get rid of it.