Representation of set valued operators

Papageorgiou, Nikolaos S.

doi:10.1090/S0002-9947-1985-0808737-3

Representation of set valued operators

Author: Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 292 (1985), 557-572
MSC: Primary 47H99; Secondary 28B20, 46E30, 46G99
DOI: https://doi.org/10.1090/S0002-9947-1985-0808737-3
MathSciNet review: 808737
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Abstract: In this paper we prove representation theorems for set valued additive operators acting on the spaces $L_X^1(X = {\text{separable Banach space)}}$ , ${L^1}$ and ${L^\infty }$ . Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be -compact in . This result is a new very general result about -compactness in the Lebesgue-Bochner space .

[1] Richard A. Alò and André de Korvin, Representation of Hammerstein operators by Nemytskii measures, J. Math. Anal. Appl. 52 (1975), no. 3, 490–513. MR 428145, https://doi.org/10.1016/0022-247X(75)90075-X
[2] Zvi Artstein, On the calculus of closed set-valued functions, Indiana Univ. Math. J. 24 (1974/75), 433–441. MR 360985, https://doi.org/10.1512/iumj.1974.24.24034
[3] Robert J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. MR 185073, https://doi.org/10.1016/0022-247X(65)90049-1
[4] Charles Castaing, Le théorème de Dunford-Pettis généralisé, Séminaire Pierre Lelong (Analyse) (année 1969), Springer, Berlin, 1970, pp. 133–151. Lecture Notes in Math., Vol. 116 (French). MR 0372612
[5] -, Un résultat de dérivation des multi-applications, Séminaire d'Analyse Convexe, University of Montpellier (1974), Exposé no. 2.
[6] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
[7] A. Costé, Set valued measures, Topology and Measure Theory (D. D. R. Zinnowitz, ed.), 1974.
[8] 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
[9] Phan Văn Chu’o’ng, On the density of extremal selections for measurable multifunctions, Acta Math. Vietnam. 6 (1981), no. 2, 13–28 (1982). MR 694272
[10] J. P. Delahaye and J. Denel, The continuities of the point-to-set maps: Definitions and equivalences, Math. Programming Stud. 10 (1979), 8-12.
[11] J. Diestel, Remarks on weak compactness in ${L^1}(,X)$ , Glasgow Math. J. 18 (1977), 87-91.
[12] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
[13] N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
[14] Henry Hermes, Calculus of set valued functions and control, J. Math. Mech. 18 (1968/1969), 47–59. MR 0231972, https://doi.org/10.1512/iumj.1969.18.18006
[15] Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121
[16] Fumio Hiai, Radon-Nikodým theorems for set-valued measures, J. Multivariate Anal. 8 (1978), no. 1, 96–118. MR 583862, https://doi.org/10.1016/0047-259X(78)90022-2
[17] Fumio Hiai, Representation of additive functionals on vector-valued normed Köthe spaces, Kodai Math. J. 2 (1979), no. 3, 300–313. MR 553237
[18] Fumio Hiai and Hisaharu Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), no. 1, 149–182. MR 507504, https://doi.org/10.1016/0047-259X(77)90037-9
[19] C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, https://doi.org/10.4064/fm-87-1-53-72
[20] 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
[21] I. M. Gelfand, Abstrakte Functionen und Lineare Operatoren, Math. Sb. 4 (1938), 235-286.
[22] Victor J. Mizel and K. Sundaresan, Representation of vector valued nonlinear functions, Trans. Amer. Math. Soc. 159 (1971), 111–127. MR 279647, https://doi.org/10.1090/S0002-9947-1971-0279647-8
[23] Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, https://doi.org/10.1016/0022-247X(71)90200-9
[24] N. S. Papageorgiou, On the theory of Banach valued multifunctions, Part 1: Integration and conditional expectation, J. Multivariate Anal. 16 (1985) (in press).
[25] R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Springer, Berlin, 1976, pp. 157–207. Lecture Notes in Math., Vol. 543. MR 0512209