= Decomposition of a BV derivative
{title2=$Du=\nabla u\,\mathcal L^n+D^ju+D^cu$}
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.
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