Integration of Correspondences on Loeb Spaces
HTML articles powered by AMS MathViewer
- by Yeneng Sun PDF
- Trans. Amer. Math. Soc. 349 (1997), 129-153 Request permission
Abstract:
We study the Bochner and Gel$^{\prime }$fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework.References
- Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR 859372
- Robert M. Anderson, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MR 654856, DOI 10.1090/S0002-9947-1982-0654856-1
- Zvi Artstein, Distributions of random sets and random selections, Israel J. Math. 46 (1983), no. 4, 313–324. MR 730347, DOI 10.1007/BF02762891
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- Robert J. Aumann, Markets with a continuum of traders, Econometrica 32 (1964), 39–50. MR 172689, DOI 10.2307/1913732
- Robert J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. MR 185073, DOI 10.1016/0022-247X(65)90049-1
- Erik J. Balder, Fatou’s lemma in infinite dimensions, J. Math. Anal. Appl. 136 (1988), no. 2, 450–465. MR 972148, DOI 10.1016/0022-247X(88)90096-0
- H. S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math. J. 5 (1939), 548–551. MR 21, DOI 10.1215/S0012-7094-39-00546-6
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Charles L. Byrne, Remarks on the set-valued integrals of Debreu and Aumann, J. Math. Anal. Appl. 62 (1978), no. 2, 243–246. MR 485918, DOI 10.1016/0022-247X(78)90123-3
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310, DOI 10.1007/BFb0087685
- 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
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Sergiu Hart, Werner Hildenbrand, and Elon Kohlberg, On equilibrium allocations as distributions on the commodity space, J. Math. Econom. 1 (1974), no. 2, 159–166. MR 436929, DOI 10.1016/0304-4068(74)90006-8
- Sergiu Hart and Elon Kohlberg, Equally distributed correspondences, J. Math. Econom. 1 (1974), no. 2, 167–174. MR 422553, DOI 10.1016/0304-4068(74)90007-X
- Fumio Hiai and Hisaharu Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), no. 1, 149–182. MR 507504, DOI 10.1016/0047-259X(77)90037-9
- Werner Hildenbrand, Core and equilibria of a large economy, Princeton Studies in Mathematical Economics, No. 5, Princeton University Press, Princeton, N.J., 1974. With an appendix to Chapter 2 by K. Hildenbrand. MR 0389160
- Bhu Dev Sharma and Ram Autar, Information-improvement functions, Econometrica 42 (1974), 103–112. MR 406699, DOI 10.2307/1913688
- Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
- Robert C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. MR 165344, DOI 10.1090/S0002-9947-1964-0165344-2
- Robert C. James, A counterexample for a $\textrm {sup}$ theorem in normed spaces, Israel J. Math. 9 (1971), 511–512. MR 279565, DOI 10.1007/BF02771466
- M. Ali Khan, On the integration of set-valued mappings in a nonreflexive Banach space. II, Simon Stevin 59 (1985), no. 3, 257–267. MR 833482
- M. Ali Khan and Mukul Majumdar, Weak sequential convergence in $L_1(\mu ,X)$ and an approximate version of Fatou’s lemma, J. Math. Anal. Appl. 114 (1986), no. 2, 569–573. MR 833611, DOI 10.1016/0022-247X(86)90108-3
- M. A. Khan and Y. N. Sun, Non-cooperative games on hyperfinite Loeb spaces, submitted.
- M. A. Khan and Y. N. Sun, General equilibrium theory with a Loeb space of agents, presented at the Workshop on Geometry, Topology and Markets at the Fields Institute for Research in Mathematical Sciences in July 1994.
- Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
- Gregory Knowles, Lyapunov vector measures, SIAM J. Control 13 (1975), 294–303. MR 0388216, DOI 10.1137/0313017
- Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
- C. Olech, Existence theory in optimal control, Control theory and topics in functional analysis (Internat. Sem., Internat. Centre Theoret. Phys., Trieste, 1974) Internat. Atomic Energy Agency, Vienna, 1976, pp. 291–328. MR 0513247
- Horst Osswald and Yeneng Sun, On the extensions of vector-valued Loeb measures, Proc. Amer. Math. Soc. 111 (1991), no. 3, 663–675. MR 1047007, DOI 10.1090/S0002-9939-1991-1047007-6
- K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
- Patrizia Pucci and Giuseppe Vitillaro, A representation theorem for Aumann integrals, J. Math. Anal. Appl. 102 (1984), no. 1, 86–101. MR 751344, DOI 10.1016/0022-247X(84)90204-X
- Salim Rashid, Economies with many agents, Johns Hopkins University Press, Baltimore, MD, 1987. An approach using nonstandard analysis. MR 871874
- Hans Richter, Verallgemeinerung eines in der Statistik benötigten Satzes der Masstheorie, Math. Ann. 150 (1963), 85–90 (German). MR 146329, DOI 10.1007/BF01396583
- Aldo Rustichini, A counterexample and an exact version of Fatou’s lemma in infinite-dimensional spaces, Arch. Math. (Basel) 52 (1989), no. 4, 357–362. MR 998411, DOI 10.1007/BF01194410
- A. Rustichini and N. Yannelis, What is perfection competition?, Equilibrium Theory in Infinite Dimensional Spaces (M. A. Khan and N. C. Yannelis, eds.), Springer-Verlag, Berlin, 1991.
- David Schmeidler, Fatou’s lemma in several dimensions, Proc. Amer. Math. Soc. 24 (1970), 300–306. MR 248316, DOI 10.1090/S0002-9939-1970-0248316-7
- Y. N. Sun, Nonstandard theory of vector measures, Ph.D. dissertation, University of Illinois, Urbana, Illinois, 1989.
- Yeneng Sun, On the theory of vector valued Loeb measures and integration, J. Funct. Anal. 104 (1992), no. 2, 327–362. MR 1153991, DOI 10.1016/0022-1236(92)90004-3
- —, Distributional properties of correspondences on Loeb spaces, Journal of Functional Analysis 139 (1996), 68–93.
- Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
- Nicholas C. Yannelis, On the upper and lower semicontinuity of the Aumann integral, J. Math. Econom. 19 (1990), no. 4, 373–389. MR 1062809, DOI 10.1016/0304-4068(90)90028-8
- —, Integration of Banach-valued correspondences, Equilibrium Theory in Infinite Dimensional Spaces (M. A. Khan and N. C. Yannelis, eds.), Springer-Verlag, Berlin, 1991.
Additional Information
- Yeneng Sun
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260
- Address at time of publication: Cowles Foundation, Yale University, New Haven, Connecticut 06520
- Email: gs53@econ.yale.edu
- Received by editor(s): February 23, 1995
- Additional Notes: The main results were presented at the Fifth Asian Logic Conference held in Singapore in June 1993. The author is grateful to Professors Robert Anderson, Donald Burkholder, Chi Tat Chong, Ward Henson, Zhuxin Hu, Jerome Keisler, Ali Khan, Peter Loeb, Walter Trockel, and Jerry Uhl for helpful conversations and encouragement. The research is partially supported by the National University of Singapore, grant no. RP3920641.
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 129-153
- MSC (1991): Primary 03H05, 28B20; Secondary 46G10, 90A14
- DOI: https://doi.org/10.1090/S0002-9947-97-01825-4
- MathSciNet review: 1401529