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 consequencesEdit

Given a field of sets   and a Banach space  , a finitely additive vector measure (or measure, for short) 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.


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, we get two different outcomes.

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

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

The variation of a vector measureEdit

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 theoremEdit

In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) finite-dimensional 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]


  1. ^ a b Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
  2. ^ a b Diestel, Joe; Uhl, Jerry J., Jr. (1977). Vector measures. Providence, R.I: American Mathematical Society. ISBN 0-8218-1515-6.
  3. ^ a b 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 0920371. OCLC 13064804.CS1 maint: ref=harv (link)
  4. ^ Roberts, John (July 1986). "Large economies". In David M. Kreps; John Roberts; Robert B. Wilson (eds.). 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.CS1 maint: ref=harv (link)
  5. ^ 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 0191623. This paper builds on two papers by Aumann:

    Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.

    Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.

  6. ^ Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. 5 (2). pp. 165–77. JSTOR 2525560.CS1 maint: ref=harv (link) 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.

  7. ^ 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 0420366.
  8. ^ a b c 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 0564562.
  9. ^ 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.
  10. ^ Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E., ed. (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5.CS1 maint: ref=harv (link)
  11. ^ 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 0514468.


See alsoEdit