A finite-dimensional vector measure has scalar measures as components and a total variation norm of a measure defined using their joint vector norm. Finite variation controls every pairing with a bounded continuous vector test. Derivatives in the bounded-variation space are examples.
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 .
Each component of a finite vector measure is absolutely continuous with respect to its variation measure. The Radon-Nikodym theorem gives a vector density . Taking the variation measure of this representation gives , hence the unit-length property. For a BV derivative this separates the magnitude of variation from its local direction.
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A vector measure is a mathematical concept that extends the idea of a measure (as found in measure theory) to a vector-valued function. In classical measure theory, a measure assigns a non-negative real number to subsets of a given space, typically based on the size or volume of those sets. In the context of vector measures, the concept is generalized to allow for values that are vectors instead of just scalars.