# 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 ${\displaystyle (\Omega ,{\mathcal {F}})}$ and a Banach space ${\displaystyle X}$, a finitely additive vector measure (or measure, for short) is a function ${\displaystyle \mu :{\mathcal {F}}\to X}$ such that for any two disjoint sets ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle {\mathcal {F}}}$ one has

${\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}$

A vector measure ${\displaystyle \mu }$ is called countably additive if for any sequence ${\displaystyle (A_{i})_{i=1}^{\infty }}$ of disjoint sets in ${\displaystyle {\mathcal {F}}}$ such that their union is in ${\displaystyle {\mathcal {F}}}$ it holds that

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i})}$

with the series on the right-hand side convergent in the norm of the Banach space ${\displaystyle X.}$

It can be proved that an additive vector measure ${\displaystyle \mu }$ is countably additive if and only if for any sequence ${\displaystyle (A_{i})_{i=1}^{\infty }}$ as above one has

${\displaystyle \lim _{n\to \infty }\left\|\mu \left(\displaystyle \bigcup _{i=n}^{\infty }A_{i}\right)\right\|=0,\quad \quad \quad (*)}$

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 ${\displaystyle [0,\infty ),}$ the set of real numbers, and the set of complex numbers.

## Examples

Consider the field of sets made up of the interval ${\displaystyle [0,1]}$ together with the family ${\displaystyle {\mathcal {F}}}$ of all Lebesgue measurable sets contained in this interval. For any such set ${\displaystyle A}$, define

${\displaystyle \mu (A)=\chi _{A}\,}$

where ${\displaystyle \chi }$ is the indicator function of ${\displaystyle A.}$ Depending on where ${\displaystyle \mu }$ is declared to take values, we get two different outcomes.

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

## The variation of a vector measure

Given a vector measure ${\displaystyle \mu :{\mathcal {F}}\to X,}$ the variation ${\displaystyle |\mu |}$ of ${\displaystyle \mu }$ is defined as

${\displaystyle |\mu |(A)=\sup \sum _{i=1}^{n}\|\mu (A_{i})\|}$

where the supremum is taken over all the partitions

${\displaystyle A=\bigcup _{i=1}^{n}A_{i}}$

The variation of ${\displaystyle \mu }$ is a finitely additive function taking values in ${\displaystyle [0,\infty ].}$ It holds that

${\displaystyle ||\mu (A)||\leq |\mu |(A)}$

for any ${\displaystyle A}$ in ${\displaystyle {\mathcal {F}}.}$ If ${\displaystyle |\mu |(\Omega )}$ is finite, the measure ${\displaystyle \mu }$ is said to be of bounded variation. One can prove that if ${\displaystyle \mu }$ is a vector measure of bounded variation, then ${\displaystyle \mu }$ is countably additive if and only if ${\displaystyle |\mu |}$ is countably additive.

## 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.[1][2][3] 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).[2] It is used in economics,[4][5][6] in ("bang–bang") control theory,[1][3][7][8] and in statistical theory.[8] Lyapunov's theorem has been proved by using the Shapley–Folkman lemma,[9] which has been viewed as a discrete analogue of Lyapunov's theorem.[8][10] [11]

## References

1. Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. {{#invoke:citation/CS1|citation |CitationClass=book }}
4. {{#invoke:citation/CS1|citation |CitationClass=book }}
5. {{#invoke:Citation/CS1|citation |CitationClass=journal }} This paper builds on two papers by Aumann:

{{#invoke:Citation/CS1|citation |CitationClass=journal }}

{{#invoke:Citation/CS1|citation |CitationClass=journal }}

6. Template:Cite news Vind's article was noted by Template:Harvtxt with this comment:

The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and if one averages those individual sets over a collection of insignificant agents, then the resulting set is necessarily convex. [Debreu appends this footnote: "On this direct consequence of a theorem of A. A. Lyapunov, see Template:Harvtxt."] But explanations of the ... functions of prices ... can be made to rest on the convexity of sets derived by that averaging process. Convexity in the commodity space obtained by aggregation over a collection of insignificant agents is an insight that economic theory owes ... to integration theory. [Italics added]

7. {{#invoke:citation/CS1|citation |CitationClass=book }}
8. Template:Cite news
9. Template:Cite news
10. {{#invoke:citation/CS1|citation |CitationClass=book }}
11. Page 210: Template:Cite news

### Books

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• Kluvánek, I., Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }}