Vector measure

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.

Definitions and first consequences

Given a field of sets and a Banach space a finitely additive vector measure is a function such that for any two disjoint sets and in one has
A vector measure is called countably additive if for any sequence of disjoint sets in such that their union is in it holds that
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 is countably additive if and only if for any sequence as above one has
where is the norm on
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 the set of real numbers, and the set of complex numbers.

Examples

Consider the field of sets made up of the interval together with the family of all Lebesgue measurable sets contained in this interval. For any such set define
where is the indicator function of Depending on where is declared to take values, two different outcomes are observed.

viewed as a function from to the -space is a vector measure which is not countably-additive.
viewed as a function from to the -space is a countably-additive vector measure.

Both of these statements follow quite easily from the criterion stated above.

The variation of a vector measure

Given a vector measure the variation of is defined as
where the supremum is taken over all the partitions
of into a finite number of disjoint sets, for all in Here, is the norm on
The variation of is a finitely additive function taking values in It holds that
for any in If is finite, the measure is said to be of bounded variation. One can prove that if is a vector measure of bounded variation, then is countably additive if and only if is countably additive.

Lyapunov's theorem

In the theory of vector measures, Lyapunov's theorem states that the range of a finite-dimensional vector measure is closed and convex. In fact, the range of a non-atomic vector measure is a zonoid. It is used in economics, in 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.