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

Papageorgiou, Nikolaos S.

doi:10.1090/S0002-9947-1987-0906815-3

Contributions to the theory of set valued functions and set valued measures
HTML articles powered by AMS MathViewer

by Nikolaos S. Papageorgiou PDF

Trans. Amer. Math. Soc. 304 (1987), 245-265 Request permission

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.

References

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), no. 4, 289–298. MR 571083, DOI 10.1007/BF02760458
James K. Brooks and N. Dinculeanu, Weak compactness in spaces of Bochner integrable functions and applications, Advances in Math. 24 (1977), no. 2, 172–188. MR 438121, DOI 10.1016/S0001-8708(77)80017-0
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
Lamberto Cesari, Convexity of the range of certain integrals, SIAM J. Control 13 (1975), 666–676. MR 0380595
Alain Costé, La propriété de Radon-Nikodým en intégration multivoque, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 22, Aii, A1515–A1518 (French, with English summary). MR 374371

Sur les multimesures a valeurs fermées bornées s’un espace de Banach

280

A. Costé and R. Pallu de la Barrière, Radon-Nikodým theorems for set-valued measures whose values are convex and closed, Comment. Math. Prace Mat. 20 (1977/78), no. 2, 283–309 (loose errata). MR 519365

Probabilities and potential

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
L. Drewnowski and W. Orlicz, Continuity and representation of orthogonally additive functionals, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 647–653 (English, with Russian summary). MR 256147
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606

Linear operators

N. Friedman and M. Katz, Additive functionals of $L_{p}$ spaces, Canadian J. Math. 18 (1966), 1264–1271. MR 206692, DOI 10.4153/CJM-1966-125-x
C. Godet-Thobie, Some results about multimeasures and their selectors, Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) Lecture Notes in Math., vol. 794, Springer, Berlin, 1980, pp. 112–116. MR 577965

Théorèmes de Radon-Nikodým multivoque et intégration par rapport à certaines multi-mesures

Fumio Hiai, Radon-Nikodým theorems for set-valued measures, J. Multivariate Anal. 8 (1978), no. 1, 96–118. MR 583862, DOI 10.1016/0047-259X(78)90022-2
Fumio Hiai, Representation of additive functionals on vector-valued normed Köthe spaces, Kodai Math. J. 2 (1979), no. 3, 300–313. MR 553237
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

The theory of analytic spaces

A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the theory of lifting, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48, Springer-Verlag New York, Inc., New York, 1969. MR 0276438
M. Ali Khan and Nikolaos S. Papageorgiou, On Cournot-Nash equilibria in generalized qualitative games with a continuum of players, Nonlinear Anal. 11 (1987), no. 6, 741–755. MR 893778, DOI 10.1016/0362-546X(87)90040-X
M. Ali Khan and Nikolaos S. Papageorgiou, On Cournot-Nash equilibria in generalized qualitative games with a continuum of players, Nonlinear Anal. 11 (1987), no. 6, 741–755. MR 893778, DOI 10.1016/0362-546X(87)90040-X
Heinz-Albrecht Klei, Sous-ensembles de $L^{1}_{E}$ sans suite-$l^{1}$, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 79–80 (French, with English summary). MR 676367
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
Pierre-Jean Laurent, Approximation et optimisation, Collection Enseignement des Sciences, No. 13, Hermann, Paris, 1972 (French). MR 0467080
Judith Reeves McKinney, Kernels of measures on completely regular spaces, Duke Math. J. 40 (1973), 915–923. MR 326362
Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, DOI 10.1016/0022-247X(71)90200-9

Propriétés de mesurabilité dans les espaces de mesures

Introduction a l’etude des multimesures

Nikolaos S. Papageorgiou, Representation of set valued operators, Trans. Amer. Math. Soc. 292 (1985), no. 2, 557–572. MR 808737, DOI 10.1090/S0002-9947-1985-0808737-3
Nikolaos S. Papageorgiou, On the theory of Banach space valued multifunctions. I. Integration and conditional expectation, J. Multivariate Anal. 17 (1985), no. 2, 185–206. MR 808276, DOI 10.1016/0047-259X(85)90078-8

On the theory of Banach space valued multifunctions. Part

Set valued martingales and set valued measures

71

Nikolaos S. Papageorgiou, On the efficiency and optimality of random allocations, J. Math. Anal. Appl. 105 (1985), no. 1, 113–135. MR 773576, DOI 10.1016/0022-247X(85)90100-3
Nikolaos S. Papageorgiou, On the efficiency and optimality of allocations. II, SIAM J. Control Optim. 24 (1986), no. 3, 452–479. MR 838050, DOI 10.1137/0324026
Nikolaos S. Papageorgiou, Efficiency and optimality in economies described by coalitions, J. Math. Anal. Appl. 116 (1986), no. 2, 497–512. MR 842816, DOI 10.1016/S0022-247X(86)80014-2
Nikolaos S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 3, 507–514. MR 840638, DOI 10.1090/S0002-9939-1986-0840638-3

Convergence theorems for Banach space valued integrable multifunctions

Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
M.-F. Sainte-Beuve, On the extension of von Neumann-Aumann’s theorem, J. Functional Analysis 17 (1974), 112–129. MR 0374364, DOI 10.1016/0022-1236(74)90008-1
M.-F. Sainte-Beuve, Some topological properties of vector measures with bounded variation and its applications, Ann. Mat. Pura Appl. (4) 116 (1978), 317–379. MR 506985, DOI 10.1007/BF02413878
Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, DOI 10.1016/0022-247X(77)90060-9
Gabriella Salinetti and Roger J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), no. 1, 18–33. MR 516381, DOI 10.1137/1021002
Michel Talagrand, Weak Cauchy sequences in $L^{1}(E)$, Amer. J. Math. 106 (1984), no. 3, 703–724. MR 745148, DOI 10.2307/2374292
Lionel Thibault, Espérances conditionnelles d’intégrandes semi-continus, Ann. Inst. H. Poincaré Sect. B (N.S.) 17 (1981), no. 4, 337–350 (French, with English summary). MR 644351
Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972. MR 0372708

Bases mathématique du calcut des probabilités