= Polar decomposition of a vector measure
{title2=$\mu=\sigma|\mu|,\quad|\sigma|=1\quad|\mu|\text{-a.e.}$}
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.
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