ABSTRACT
The mean divergence of a vector field is the density of its flux. For controlled vector fields the mean and ordinary divergence coincide almost everywhere. In general, the mean divergence may exist even if the ordinary divergence does not. Viewing the flux of a vector field as an additive function of dyadic figures, we give a sufficient condition under which the mean divergence exists and determines the flux. This is accomplished by replacing the classical variation of an additive function by a suitable Borel measure — the idea originally introduced by B.S. Thomson [73] in the real line. Throughout this chapter, Ω is a fixed open subset of https://www.w3.org/1998/Math/MathML"> R n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq6919.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
