# Surface measure for Lipschitz domain

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.

-

## 1 Answer

Let $\sigma$ be the restriction of the $(d-1)$-dimensional Hausdorff measure to $\partial D$. Since $D$ is bounded with Lipschitz boundary, $\sigma$ is a finite measure. The Krein-Milman theorem implies that $\sigma$ is a weak*-limit of sums of point masses: that is, there exists a sequence $\nu_k$ of finite combinations of point masses (with nonnegative coefficients and total mass equal to $\sigma(\mathbb R^d)$ ) such that $\int g\,d\nu_k\to \int g\,d\sigma$ for every function $g\in C(\mathbb R^d)$. This yields the desired convergence.

Putting the measures $\nu_k$ into the desired form, even with $\Phi\equiv 1$, should be straightforward: take the mass of each point to be a rational number, hide their common denominator in $\epsilon^{d-1}$, and split integer masses into tightly bunched unit masses.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 26, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.915665328502655, "perplexity_flag": "head"}