A General Theorem of Gau{\ss} Using Pure Measures

TL;DR

This paper develops a general divergence theorem framework involving pure measures, establishing existence results for normal measures on finite perimeter sets and extending Gauss's theorem to unbounded vector fields with divergence measure.

Contribution

It introduces a unified approach to divergence theorems using finitely additive pure measures, generalizing classical results to broader classes of vector fields and domains.

Findings

Existence of pure normal measures for finite perimeter sets

Extension of Gauss's theorem to unbounded vector fields with divergence measure

Pure measures act on the gradient of scalar fields with core on the boundary

Abstract

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield a Gau{\ss} formula for essentially bounded vector fields having divergence measure. The second extends a result of Silhavy [19] on normal traces. In particular, it is shown that a Gau{\ss} Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field. All of these measures are shown to have their core on the boundary of the domain of integration.