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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] Robert J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. MR 0185073, https://doi.org/10.1016/0022-247X(65)90049-1
[2] Robert J. Aumann, Existence of competitive equilibria in markets with a continuum of traders, Econometrica 34 (1966), 1–17. MR 0191623, https://doi.org/10.2307/1909854
[3] H. T. Banks and Marc Q. Jacobs, A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246–272. MR 0265937, https://doi.org/10.1016/0022-247X(70)90078-8
[4] T. F. Bridgland Jr., Trajectory integrals of set valued functions, Pacific J. Math. 33 (1970), 43–68. MR 0262454
[5] Ch. Castaing, Sur les multi-applications mesurables, Rev. Française Informat. Recherche Opérationnell 1 (1967), no. 1, 91–126 (French). MR 0223527
[6] Richard Datko, Measurability properties of set-valued mappings in a Banach space, SIAM J. Control 8 (1970), 226–238. MR 0274691
[7] Gerard Debreu, Integration of correspondences, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 351–372. MR 0228252
[8] Gerard Debreu and David Schmeidler, The Radon-Nikodým derivative of a correspondence, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 41–56. MR 0407230
[9] Hubert Halkin, Some further generalizations of a theorem of Lyapounov, Arch. Rational Mech. Anal. 17 (1964), 272–277. MR 0174421, https://doi.org/10.1007/BF00282290
[10] Paul R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948), 416–421. MR 0024963, https://doi.org/10.1090/S0002-9904-1948-09020-6
[11] Henry Hermes, Calculus of set valued functions and control, J. Math. Mech. 18 (1968/1969), 47–59. MR 0231972
[12] Henry Hermes and Joseph P. LaSalle, Functional analysis and time optimal control, Academic Press, New York-London, 1969. Mathematics in Science and Engineering, Vol. 56. MR 0420366
[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 0236341, https://doi.org/10.4153/CJM-1969-041-7
[15] Masuo Hukuhara, Intégration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223 (French). MR 0226503
[16] Marc Q. Jacobs, On the approximation of integrals of multivalued functions, SIAM J. Control 7 (1969), 158–177. MR 0254209
[17] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970.
[18] R. T. Rockafellar, Measurable dependence of convex sets and functions on parameters, J. Math. Anal. Appl. 28 (1969), 4–25. MR 0247019, https://doi.org/10.1016/0022-247X(69)90104-8
[19] David Schmeidler, Convexity and compactness in countably additive correspondences, Differential Games and Related Topics (Proc. Internat. Summer School, Varenna, 1970) North-Holland, Amsterdam, 1971, pp. 235–242. MR 0306442
[20] K. Vind, Edgeworth-allocations in an exchange economy with many traders, Int. Econ. Rev. 5 (1964), 165-177.