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

\mu(A\cup B) =\mu(A) + \mu (B).

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

\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 $X.$

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

\lim_{n\to\infty}\left\|\mu\left(\displaystyle\bigcup_{i=n}^\infty A_i\right)\right\|=0, \quad\quad\quad (*)

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

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

Examples

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

\mu(A)=\chi_A\,

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

$\mu,$ viewed as a function from $\mathcal F$ to the L^p-space $L^\infty([0, 1]),$ is a vector measure which is not countably-additive.
$\mu,$ viewed as a function from $\mathcal F$ to the L^p-space $L^1([0, 1]),$ 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 $\mu:\mathcal{F}\to X,$ the variation $|\mu|$ of $\mu$ is defined as

|\mu|(A)=\sup \sum_{i=1}^n \|\mu(A_i)\|

where the supremum is taken over all the partitions

A=\bigcup_{i=1}^n A_i

of $A$ into a finite number of disjoint sets, for all $A$ in $\mathcal{F}$ . Here, $\|\cdot\|$ is the norm on $X.$

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

||\mu(A)||\le |\mu|(A)

for any $A$ in $\mathcal{F}.$ If $|\mu|(\Omega)$ is finite, the measure $\mu$ is said to be of bounded variation. One can prove that if $\mu$ is a vector measure of bounded variation, then $\mu$ is countably additive if and only if $|\mu|$ 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 2 Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
1 2 Diestel, Joe; Uhl, Jerry J., Jr. (1977). Vector measures. Providence, R.I: American Mathematical Society. ISBN 0-8218-1515-6.
1 2 Rolewicz, Stefan (1987). Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series) 29 (Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers. pp. xvi+524. ISBN 90-277-2186-6. MR 920371. OCLC 13064804
↑ Roberts, John (July 1986). "Large economies". In David M. Kreps; John Roberts; Robert B. Wilson. Contributions to the New Palgrave (PDF). Research paper 892. Palo Alto, CA: Graduate School of Business, Stanford University. pp. 30–35. (Draft of articles for the first edition of New Palgrave Dictionary of Economics). Retrieved 7 February 2011
↑ Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 191623. This paper builds on two papers by Aumann:
"Markets with a continuum of traders". Econometrica 32 (1–2): 39–50. January–April 1964. doi:10.2307/1913732. JSTOR 1913732. MR 172689.

"Integrals of set-valued functions". Journal of Mathematical Analysis and Applications 12 (1): 1–12. August 1965. doi:10.1016/0022-247X(65)90049-1. MR 185073.
↑ Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review 5 (2). pp. 165–77. JSTOR 2525560. Vind's article was noted by Debreu (1991, p. 4) 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 Vind (1964)."] 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]
Debreu, Gérard (March 1991). "The Mathematization of economic theory". The American Economic Review. 81, number 1 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC). pp. 1–7. JSTOR 2006785.
↑ Hermes, Henry; LaSalle, Joseph P. (1969). Functional analysis and time optimal control. Mathematics in Science and Engineering 56. New York—London: Academic Press. pp. viii+136. MR 420366.
1 2 3 Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review 22 (2). pp. 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 564562.
↑ Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem". SIAM Journal on Control and Optimization 28 (2). pp. 478–481. doi:10.1137/0328026. MR 1040471.
↑ Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E., ed. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi:10.1057/9780230226203.1518.
↑ Page 210: Mas-Colell, Andreu (1978). "A note on the core equivalence theorem: How many blocking coalitions are there?". Journal of Mathematical Economics 5 (3). pp. 207–215. doi:10.1016/0304-4068(78)90010-1. MR 514468.

Books

Cohn, Donald L. (1997) [1980]. Measure theory (reprint ed.). Boston–Basel–Stuttgart: Birkhäuser Verlag. pp. IX+373. ISBN 3-7643-3003-1. Zbl 0436.28001
Diestel, Joe; Uhl, Jerry J., Jr. (1977). Vector measures. Mathematical Surveys 15. Providence, R.I: American Mathematical Society. pp. xiii+322. ISBN 0-8218-1515-6.
Kluvánek, I., Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
van Dulst, D. (2001), "Vector measures", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Rudin, W (1973). Functional analysis. New York: McGraw-Hill. p. 114.