Kluvánek introduced the concept of a closed vector measure.

In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties.

In other words, the limit of a sequence of vector measures is a vector measure.

The normalized magnitude of the residual vector measures how well the can be constructed from the k columns of B .

As noted for the case of general functions of bounded variation, this vector measure is the distributional or weak gradient of .

The notion of a closed vector measure stimulated much research, especially by W. Graves and his students at Chapel Hill, North Carolina.

This is due to , and is often used to define the integral with respect to vector measures , and especially vector-valued Radon measures.

In advanced measure-theory, the Shapley-Folkman lemma has been used to prove Lyapunov's theorem, which states that the range of a vector measure is convex.

In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) vector measure is closed and convex.

One can prove that if is a vector measure of bounded variation, then is countably additive if and only if is countably additive.

A probability measure is a finite measure, and the Shapley-Folkman lemma has applications in non-probabilistic measure theory, such as the theories of volume and of vector measures.

This notion was crucial for his investigations of the range of a vector measure and led to the extension to infinite dimensional spaces of the classical Liapunov convexity theorem, together with many consequences and applications.

As successful as the theory of integration with respect to countably additive vector measures has been in various branches of mathematics, such as mathematical physics, functional analysis and operator theory, for example, it is also known that there are fundamental problems which cannot be treated in this way.

Countably additive vector measures defined on sigma-algebras are more general than measures, signed measures, and complex measures, which are countably additive functions taking values respectively on the extended interval the set of real numbers, and the set of complex numbers.

Whereas the Laplace-Stieltjes transform of a real-valued function is a special case of the Laplace transform of a measure applied to the associated Stieltjes measure, the conventional Laplace transform cannot handle vector measures: measures with values in a Banach space.