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 (or measure such that for any two disjoint sets

is called countably additive if for any sequence with the series on the right-hand side convergent in the norm

as above one has

Countably additive vector measures defined on sigma-algebras are more general than finite measuress, finite signed measures, and complex measures, which are countably additive functions the set of real numbers, and the set of complex numbers.

Examples

of all Lebesgue measurable sets is the indicator function is declared to take values, two different outcomes are observed.

• to the L^p-space is a vector measure which is not countably-additive.

• is a countably-additive vector measure.

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

The variation of a vector measure

the variation where the supremum is taken over all the partitions

is said to be of bounded variation is countably additive.

Lyapunov's theorem

In the theory of vector measures, Lyapunov states that the range of a (non-atomic) finite-dimensional vectormeasure 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.

See also

References

Bibliography

• Kluvánek, I., Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies20, Amsterdam, 1976.