In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
Contents
Definitions and first consequences
Given a field of sets
A vector measure
with the series on the right-hand side convergent in the norm of the Banach space
It can be proved that an additive vector measure
where
Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval
Examples
Consider the field of sets made up of the interval
where
Both of these statements follow quite easily from the criterion (*) stated above.
The variation of a vector measure
Given a vector measure
where the supremum is taken over all the partitions
of
The variation of
for any
Lyapunov's theorem
In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) vector measure is closed and convex. In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes). It is used in economics, in ("bang–bang") control theory, and in statistical theory. Lyapunov's theorem has been proved by using the Shapley–Folkman lemma, which has been viewed as a discrete analogue of Lyapunov's theorem.