Decomposition of a BV derivative ($Du=\nabla u{\mathcal{L}}^n+D^{j}u+D^{c}u$)

The Lebesgue decomposition theorem separates the absolutely continuous part from the singular part. BV fine structure then splits the latter into the jump part of a BV derivative and the Cantor part of a BV derivative. The density of the absolutely continuous part is the almost-everywhere approximate gradient. These three mutually singular components distinguish smooth variation, surface discontinuities and diffuse singular variation.

This derivative component is carried by the jump set of a BV function. Swapping the two BV traces on a hypersurface and reversing the normal leaves it unchanged. In one dimension it consists of the point measure atoms, each weighted by the difference between the right and left limits.