Set-valued measures

Artstein, Zvi

doi:10.1090/S0002-9947-1972-0293054-4

Set-valued measures

Author: Zvi Artstein
Journal: Trans. Amer. Math. Soc. 165 (1972), 103-125
MSC: Primary 28A45
DOI: https://doi.org/10.1090/S0002-9947-1972-0293054-4
MathSciNet review: 0293054
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Abstract: A set-valued measure is a $\sigma$ -additive set-function which takes on values in the nonempty subsets of a euclidean space. It is shown that a bounded and non-atomic set-valued measure has convex values. Also the existence of selectors (vector-valued measures) is investigated. The Radon-Nikodym derivative of a set-valued measure is a set-valued function. A general theorem on the existence of R.-N. derivatives is established. The techniques require investigations of measurable set-valued functions and their support functions.

[1] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. MR 32 #2543. MR 0185073 (32:2543)
[2] -, Existence of competitive equilibria in markets with a continuum of traders, Econometrica 34 (1966), 1-17. MR 32 #9026. MR 0191623 (32:9026)
[3] H. T. Banks and M. Q. Jacobs, A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246-272. MR 42 #846. MR 0265937 (42:846)
[4] T. F. Bridgland, Trajectory integrals of set valued functions, Pacific J. Math. 33 (1970), 43-68. MR 41 #7061. MR 0262454 (41:7061)
[5] C. Castaing, Sur les multi-applications measurables, Rev. Française Informat. Recherche Opérationnelle 1 (1967), 3-34. MR 0223527 (36:6575)
[6] R. Datko, Measurability properties of set-valued mapping in a Banach space, SIAM J. Control 8 (1970), 226-238. MR 0274691 (43:453)
[7] G. Debreu, Integration of correspondences, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), vol. II: Contributions to Probability Theory, part 1, Univ. California Press, Berkeley, Calif., 1967, pp. 351-372. MR 37 #3835. MR 0228252 (37:3835)
[8] G. Debreu and D. Schmeidler, The Radon-Nikodym derivative of a correspondence, Proc. Sixth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1970) (to appear). MR 0407230 (53:11013)
[9] H. Halkin, Some further generalizations of a theorem of Lyapunov, Arch. Rational Mech. Anal. 17 (1964), 272-277. MR 30 #4625. MR 0174421 (30:4625)
[10] P. R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948), 416-421. MR 9, 574. MR 0024963 (9:574h)
[11] H. Hermes, Calculus of set valued functions and control, J. Math. Mech. 18 (1968/69), 47-59. MR 38 #298. MR 0231972 (38:298)
[12] H. Hermes and J. P. Lasalle, Functional analysis and time optimal control, Academic Press, New York, 1969. MR 0420366 (54:8380)
[13] W. Hildenbrand, Pareto optimality for a measure space of economic agents, Int. Econ. Rev. 10 (1969), 363-372.
[14] C. J. Himmelberg and F. S. Van Vleck, Some selection theorems for measurable functions, Canad. J. Math. 21 (1969), 394-399. MR 38 #4637. MR 0236341 (38:4637)
[15] M. Hukuhara, Intégration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205-223. MR 37 #2092. MR 0226503 (37:2092)
[16] M. Q. Jacobs, On the approximation of integrals of multivalued functions, SIAM J. Control 7 (1969), 158-177. MR 40 #7419. MR 0254209 (40:7419)
[17] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970.
[18] -, Measurable dependence of convex sets and functions on parameters, J. Math. Anal. Appl. 28 (1969), 4-25. MR 40 #288. MR 0247019 (40:288)
[19] D. Schmeidler, ``Convexity and compactness in countably additive correspondence'' in Differential games and related topics, North-Holland, Amsterdam, 1971. MR 0306442 (46:5568)
[20] K. Vind, Edgeworth-allocations in an exchange economy with many traders, Int. Econ. Rev. 5 (1964), 165-177.