Vector Radon measure ($\mathcal{M}(\Omega ;{\mathbb{R}}^n)$)

A vector Radon measure has Radon measures as components. The vector form of the Riesz-Markov-Kakutani representation theorem identifies finite such measures with bounded functionals on $C_0(\Omega ;{\mathbb{R}}^n)$.

Each component of a finite vector measure is absolutely continuous with respect to its variation measure. The Radon-Nikodym theorem gives a vector density $\sigma$. Taking the variation measure of this representation gives $|\mu |=|\sigma ||\mu |$, hence the unit-length property. For a BV derivative this separates the magnitude of variation from its local direction.