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Contributions to the theory of set valued functions and set valued measures


Author: Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 304 (1987), 245-265
MSC: Primary 28B20; Secondary 46G10, 49A50, 54C60, 90A14
DOI: https://doi.org/10.1090/S0002-9947-1987-0906815-3
MathSciNet review: 906815
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Abstract: Measurable multifunctions and multimeasures with values in a Banach space are studied. We start by proving a variation of the known Dunford theorem for weak compactness in $ {L^1}(X)$. With a similar technique we prove that the range of certain vector valued integrals that appear in applications is $ w$-compact and convex. Also we obtain Dunford-Pettis type theorems for sequences of integrably bounded multifunctions. Some pointwise $ w$-compactness theorems are also obtained for certain families of measurable multifunctions. Then we prove a representation theorem for additive, set valued operators defined on $ {L^1}(X)$. Finally, in the last section, a detailed study of transition multimeasures is conducted and several representation theorems are proved.


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  • [1] J. Bourgain, An averaging result for $ {l^1}$-sequences and applications to weakly conditionally compact sets in $ {L^1}(X)$, Israel J. Math. 32 (1979), 289-299. MR 571083 (81i:46021)
  • [2] J. Brooks and N. Dinculeanu, Weak compactness in spaces of Bochner integrable functions and applications, Adv. in Math. 24 (1977), 172-188. MR 0438121 (55:11040)
  • [3] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin and New York, 1977. MR 0467310 (57:7169)
  • [4] L. Cesari, Convexity of the range of certain integrals, SIAM J. Control 13 (1975), 666-676. MR 0380595 (52:1494)
  • [5] A. Costé, La propriété de Radon-Nikodym en intégration multivoque, C. R. Acad. Sci. Paris 280 (1975), 1515-1518. MR 0374371 (51:10571)
  • [6] -, Sur les multimesures a valeurs fermées bornées s'un espace de Banach, C. R. Acad. Sci. Paris 280 (1975), 567-570.
  • [7] A. Costé and R. Pallu de la Barrière, Radon-Nikodym theorems for set valued measures whose values are convex and closed, Ann. Soc. Math. Polon. Series I: Comm. Math. 20 (1978), 283-309. MR 519365 (80h:28015)
  • [8] C. Dellacherie and A. Meyer, Probabilities and potential, Math. Studies, vol. 29, North-Holland, Amsterdam, 1978.
  • [9] J. Diestel and J. Uhl, Vector measures, Math. Surveys, vol. 15, Amer. Math. Soc., Providence, R.I., 1977. MR 0453964 (56:12216)
  • [10] L. Drewnowski and W. Orlicz, Continuity and representation of orthogonally additive functionals, Bull. Acad. Polon. Sci. Math. 17 (1969), 647-653. MR 0256147 (41:806)
  • [11] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [12] N. Dunford and J. Schwartz, Linear operators, Wiley, New York, 1958.
  • [13] N. A. Friedman and M. Katz, Additive functionals on $ {L_p}$-spaces, Canad. J. Math. 18 (1966), 1264-1271. MR 0206692 (34:6510)
  • [14] C. Godet-Thobie, Some results about multimeasures and their selectors, Measure Theory at Oberwolfach 1979 (Kolzow, ed.), Lecture Notes in Math., vol. 794, Springer-Verlag, Berlin, 1977, pp. 112-116. MR 577965 (81g:28011)
  • [15] -, Théorèmes de Radon-Nikodým multivoque et intégration par rapport à certaines multi-mesures, Séminaire d'Analyse Convexe, Exposé no. 11, Montpellier, 1974.
  • [16] F. Hiai, Radon-Nikodym theorems for set valued measures, J. Multivariate Anal. 8 (1978), 96-118. MR 0583862 (58:28411)
  • [17] -, Representation of additive functionals on vector valued normed Köthe spaces, Kodai Math. J. 2 (1979), pp. 303-313. MR 553237 (81d:46037)
  • [18] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182. MR 0507504 (58:22463)
  • [19] W. Hildenbrand, Core and equilibria of a large economy, Princeton Univ. Press, Princeton, N. J., 1974. MR 0389160 (52:9991)
  • [20] J. Hoffman-Jørgensen, The theory of analytic spaces, Publ. no. 10, Aarhus Univ., 1970.
  • [21] A. Ionescu-Tulcea and C. Ionescu-Tulcea, Topics in the theory of lifting, Springer-Verlag, Berlin, 1969. MR 0276438 (43:2185)
  • [22] M. A. Khan and N. S. Papageorgiou, Cournot-Nash equilibria for generalized qualitative games with a continuum of players, Nonlinear Anal. T.M.A. (to appear). MR 893778 (88h:90042a)
  • [23] -, Cournot-Nash equilibria in generalized qualitative games with an atomless measure space of agents, Proc. Amer. Math. Soc. 100 (1987), 505-510. MR 891154 (88h:90042b)
  • [24] H. A. Klei, Sous-ensembles de $ {L^1}(E)$ sans suite-$ {l^1}$, C. R. Acad. Sci. Paris 295 (1982), 79-80. MR 676367 (83j:46032)
  • [25] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1985. MR 752692 (86a:90012)
  • [26] P.-J. Laurent, Approximation and optimisation, Hermann, Paris, 1972. MR 0467080 (57:6947)
  • [27] J. McKinney, Kernels of measures on completely regular spaces, Duke Math. J. 40 (1973), 915-923. MR 0326362 (48:4706)
  • [28] U. Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 0283586 (44:817)
  • [29] M. F. Nouguès-Saint-Beuve, Propriétés de mesurabilité dans les espaces de mesures, Séminaire d'Analyse Convexe, Exposé no. 5, Montpellier, 1982.
  • [30] R. Pallu de la Barrière, Introduction a l'etude des multimesures, Seminaire d'Initiation a l'Analyse (G. Choquet et. al., eds.), 19e année, no. 7, 1979/80.
  • [31] N. S. Papageorgiou, Representation of set valued operators, Trans. Amer. Math. Soc. 292 (1985), 557-572. MR 808737 (87c:47085)
  • [32] -, On the theory of Banach space valued multifunctions. Part 1: Integration and conditional expectation, J. Multivariate Anal. 17 (1985), 185-206. MR 808276 (87e:28015)
  • [33] -, On the theory of Banach space valued multifunctions. Part 2: Set valued martingales and set valued measures, J. Multivariate Anal. 71 (1985), 207-227.
  • [34] -, On the efficiency and optimality of random allocations, J. Math. Anal. Appl. 105 (1985), 113-136. MR 773576 (86i:90018)
  • [35] -, On the efficiency and optimality of allocations. II, SIAM J. Control Optim. 24 (1986), 452-479. MR 838050 (87j:90035)
  • [36] -, Efficiency and optimality in economies described by coalitions, J. Math. Anal. Appl. 116 (1986), 497-512. MR 842816 (87i:90070)
  • [37] -, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 507-514. MR 840638 (88a:60117)
  • [38] -, Convergence theorems for Banach space valued integrable multifunctions, Internat. J. Math. Math. Sci. (in press).
  • [39] H. Rosenthal, A characterization of Banach spaces containing $ {l^1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 0358307 (50:10773)
  • [40] M.-F. Saint-Beuve, On the extension of Von Neumann-Aumann's theorem, J. Funct. Anal. 17 (1974), 112-129. MR 0374364 (51:10564)
  • [41] -, Some topological properties of vector measures of bounded variation and its applications, Ann. Mat. Pura Appl. 66 (1978), 317-379. MR 506985 (81d:46050)
  • [42] G. Salinetti and R. Wets, On the relation between two types of convergence of convex functions, J. Math. Anal. Appl. 60 (1977), 211-226. MR 0479398 (57:18828)
  • [43] -, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), 18-33. MR 516381 (80h:52007)
  • [44] M. Talagrand, Weak Cauchy sequences in $ {L^1}(E)$, Amer. J. Math. 106 (1984), 703-724. MR 745148 (85j:46062)
  • [45] L. Thibault, Esperances conditionelles d'integrandes semi-continues, Ann. Inst. H. Poincaré Ser. B 17 (1981), 337-350. MR 644351 (83a:60005)
  • [46] D. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-907. MR 0486391 (58:6137)
  • [47] J. Warga, Optimal control of differential and functional equations, Academic Press, New York, 1970. MR 0372708 (51:8915)
  • [48] J. Neveu, Bases mathématique du calcut des probabilités, Masson, Paris, 1964.

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DOI: https://doi.org/10.1090/S0002-9947-1987-0906815-3
Article copyright: © Copyright 1987 American Mathematical Society

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