Polar decomposition of a vector measure ($\mu =\sigma |\mu |,|\sigma |=1|\mu |\text{-a.e.}$)

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.