Transactions of the American Mathematical Society

Author(s): Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 349 (1997), 1421-1436.
MSC (1991): Primary 47H30, 49N15
Retrieve article in: PDF
This article is available free of charge

Abstract: We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their $\varepsilon$ -subdifferential $(\varepsilon \ge 0)$ and their $\varepsilon$ -directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.

1.: R. Aumann, Markets with a continuum of traders, Econometrica 32 (1964), 39-50. MR 30:2908
2.: -, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. MR 32:2543
3.: R. Aumann and L. Shapley, Values of Nonatomic Games, Princeton Univ. Press, Princeton, 1974.
4.: J.-M. Bismut, Integrales convexes et probabilités, J. Math. Anal. Appl. 42 (1973), 639-673. MR 48:3083
5.: N. Bourbaki, Espaces Vectoriels Topologiques, Hermann, Paris, 1966. MR 34:3277
6.: C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, Berlin, 1977. MR 57:7169
7.: J. Diestel and J. Uhl, Vector Measures, Math Surveys, Vol. 15, AMS, Providence, RI, 1977. MR 56:12216
8.: F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multiv. Anal. 7 (1977), 149-182. MR 58:22463
9.: J.-B. Hiriart-Urruty, Thèse d'Etat, Chapter IV, Clermond-Ferrand, France, 1977.
10.: -, Lipschitz -continuity of the approximate subdifferential of a convex function, Math Scand. 47 (1980), 123-134. MR 82c:58007
11.: A. Ioffe and V. Levin, Subdifferentials and convex functions, Trans. Moscow Math. Soc. 26 (1972), 1-72. MR 51:8817
12.: A. Ioffe and V. Tihomirov, The duality of convex functions and extremal problems, Russian Math. Surveys 23 (1968), 53-124. MR 44:5797
13.: -, Theory of Extremal Problems, North-Holland, Amsterdam, 1979. MR 80d:49001b
14.: H.-A. Klei, Thèse d'Etat, Université de Paris VI, Paris, 1985.
15.: N. Komuro, Basic properties of convex functions and convex integrands, Hokkaido Math. Jour. 18 (1989), 1-30. MR 90f:49008
16.: P.-J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972. MR 57:6947
17.: V. Levin, The Lebesgue decomposition for functionals on the vector-function space $L^\infty _X$ , Functional Anal. Appl. 8 (1974), 314-317. MR 57:850
18.: -, Convex integral functionals and the theory of lifting, Russian Math. Surveys 30 (1975), 115-178. MR 53:3698
19.: N. S. Papageorgiou, Convergence and representation theorems for set-valued random processes, J. Math. Anal. Appl. 150 (1990), 129-145. MR 92a:60117
20.: -, Weak convergence of random sets in Banach spaces, J. Math. Anal. Appl. 164 (1992), 571-589. MR 92m:49013
21.: R. T. Rockafellar, Integrals which are convex functionals I, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
22.: -, Integrals which are convex functionals II, Pacific J. Math. 39 (1971), 439-469. MR 46:9710
23.: -, Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis, ed. by E. Zarantonello, Academic Press, New York, 1971, pp. 215-236. MR 52:11693
24.: -, Conjugate Duality and Optimization, Reg. Conf. Series in Appl. Math., vol. 16, SIAM, Philadelphia, 1973. MR 51:9811
25.: -, Integral functionals, normal integrands and measurable multifunctions, in Nonlinear Operators and the Calculus of Variations, ed. by J.-P. Gossez et al., Springer, New York, 1976, pp. 157-207. MR 58:23598
26.: -, Convex Analysis, Princeton Univ. Press, Princeton, 1970. MR 43:445
27.: R. T. Rockafellar and R. J.-B. Wets, On the interchange of subdifferentiation and conditional expectation for convex functionals, Stochastics 10 (1982), 173-182. MR 83j:49022
28.: M.-F. Saint-Beuve, On the extension of von Neumann-Aumann theorem, J. Funct. Anal. 17 (1974), 112-129. MR 51:10564
29.: L. Thibault, Esperance conditionelle d'integrandes semicontinus, Ann. Inst. H. Poincare 17 (1981), 337-350. MR 83a:60005
30.: M. Valadier, Integration de convexes fermes notamment d'epigraphes et $\inf$ -convolution continue, RAIRO 4 (1970), 57-73. MR 43:3881
31.: D. Wagner, Survey of measurable selection theorems, SIAM J. Control. Optim. 15 (1977), 859-903. MR 58:6137

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H30, 49N15

Nikolaos S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

DOI: 10.1090/S0002-9947-97-01478-5
PII: S 0002-9947(97)01478-5
Keywords: Normal convex integrand, $\varepsilon$-subdifferential, multifunction, support function, singular functional, conditional expectation, Souslin space subdifferential
Received by editor(s): January 13, 1994
Received by editor(s) in revised form: January 23, 1995
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS