Vector measure

 ( Control Theory ) 

In mathematics, a vector measure is a function defined on a family of sets and taking vector space values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real number values only.

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Definitions and first consequences Given a field of sets $(\backslash Omega,\; \backslash mathcal\; F)$ and a Banach space $X,$ a finitely additive vector measure (or measure, for short) is a function $\backslash mu:\backslash mathcal\; \{F\}\; \backslash to\; X$ such that for any two $A$ and $B$ in $\backslash mathcal\{F\}$ one has $$\backslash mu(A\backslash cup\; B)\; =\backslash mu(A)\; +\; \backslash mu\; (B).$$

A vector measure $\backslash mu$ is called countably additive if for any sequence $(A\_i)\_\{i=1\}^\{\backslash infty\}$ of disjoint sets in $\backslash mathcal\; F$ such that their union is in $\backslash mathcal\; F$ it holds that $$\backslash mu\{\backslash left(\backslash bigcup\_\{i=1\}^\backslash infty\; A\_i\backslash right)\}\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\backslash mu(A\_i)$$ with the series on the right-hand side convergent in the norm of the Banach space $X.$

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

where $\backslash |\backslash cdot\backslash |$ is the norm on $X.$

Countably additive vector measures defined on are more general than finite measures, finite , and , which are countably additive functions taking values respectively on the real interval $[0,\; \backslash infty),$ the set of , and the set of .

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Examples Consider the field of sets made up of the interval $0,$ together with the family $\backslash mathcal\; F$ of all Lebesgue measurable sets contained in this interval. For any such set $A,$ define $$\backslash mu(A)\; =\; \backslash chi\_A$$ where $\backslash chi$ is the indicator function of $A.$ Depending on where $\backslash mu$ is declared to take values, two different outcomes are observed.

• $\backslash mu,$ viewed as a function from $\backslash mathcal\; F$ to the Lp space $L^\backslash infty(0,),$ is a vector measure which is not countably-additive.
• $\backslash mu,$ viewed as a function from $\backslash mathcal\; F$ to the $L^p$-space $L^1(0,),$ is a countably-additive vector measure.

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

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The variation of a vector measure Given a vector measure $\backslash mu\; :\; \backslash mathcal\{F\}\; \backslash to\; X,$ the variation $|\backslash mu|$ of $\backslash mu$ is defined as $$|\backslash mu|(A)=\backslash sup\; \backslash sum\_\{i=1\}^n\; \backslash |\backslash mu(A\_i)\backslash |$$ where the supremum is taken over all the partitions $$A\; =\; \backslash bigcup\_\{i=1\}^n\; A\_i$$ of $A$ into a finite number of disjoint sets, for all $A$ in $\backslash mathcal\{F\}.$ Here, $\backslash |\backslash cdot\backslash |$ is the norm on $X.$

The variation of $\backslash mu$ is a finitely additive function taking values in $0,.$ It holds that $$\backslash |\backslash mu(A)\backslash |\; \backslash leq\; |\backslash mu|(A)$$ for any $A$ in $\backslash mathcal\{F\}.$ If $|\backslash mu|(\backslash Omega)$ is finite, the measure $\backslash mu$ is said to be of bounded variation. One can prove that if $\backslash mu$ is a vector measure of bounded variation, then $\backslash mu$ is countably additive if and only if $|\backslash mu|$ is countably additive.

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Lyapunov's theorem In the theory of vector measures, Alexey Lyapunov* states that the range of a (non-atomic) finite-dimensional vector measure is closed set and convex set.Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies  20, Amsterdam, 1976.

(1977). 9780821815151, American Mathematical Society. ISBN 9780821815151

Stefan Rolewicz (1987). 9789027721860, D. Reidel Publishing Co.; PWN—Polish Scientific Publishers. ISBN 9789027721860

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 ). It is used in economics, This paper builds on two papers by Aumann:

Vind's article was noted by 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 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

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 discretization analogue of Lyapunov's theorem.

Ross M. Starr (2025). 9780333786765, Palgrave Macmillan. . ISBN 9780333786765

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See also

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Bibliography

• Donald L. Cohn (1997). 9783764330033, Birkhäuser Verlag. . ISBN 9783764330033
• (1977). 9780821815151, American Mathematical Society. ISBN 9780821815151
• Kluvánek, I., Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies  20, Amsterdam, 1976.
• (1973). 9780070542259, McGraw-Hill. . ISBN 9780070542259

Categories: Control Theory, Functional Analysis, Measures

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