# zbMATH — the first resource for mathematics

A characterization theorem for normal integrands with applications to descriptive function theory, functional analysis and nonconvex optimization. (English) Zbl 0834.54007
A real function $$f$$ of two variables which is measurable in the first variable $$\omega$$ on a measurable space $$\Omega$$ and continuous with respect to the other variable $$x$$ on a separable metric space $$X$$ is called a Carathéodory integrand. Using the Novikov-Castaing representation, one gets that such function is a normal integrand. The latter means that $$f$$ is l.s.c. in $$x$$ and the set-valued map $$\text{epi } f: \omega\mapsto$$ epigraph of $$f(\omega, \cdot)$$ is measurable. The main theorem gives several characterizations of normal integrands, in particular a representation of normal integrands as suprema of Carathéodory ones. This result was announced in [the author, Sov. Math., Dokl. 21, 771-775 (1980); translation from Dokl. Akad. Nauk SSSR 252, 535-539 (1980; Zbl 0483.28013)] and is proved here even for $$X$$ being metrizable and $$\sigma$$-compact. This was also proved by A. Kucia and A. Nowak [Mat. Metody Sots. Naukakh 22, 29-33 (1989; Zbl 0742.49009)].
One of direct corollaries says that each function $$f$$ which is u.s.c. in the first variable $$\omega\in \Omega$$ (for $$\Omega$$ topological) and l.s.c. in the second variable $$x\in X$$ for $$X$$ being the countable union of some compact metrizable spaces is jointly Borel measurable.
Another application deals with integration of normal integrands and gives a possibility to use a limit procedure using the representation by a sequence of Carathéodory integrands ensured by the main theorem.
The nonconvex optimization problem concerns minimization of a functional defined by integration of a normal integrand with finitely many constraints given in terms of integrals of some normal and some Carathéodory integrands. The existence of the solution of such a problem is proved.
Reviewer: P.Holicky (Praha)
##### MSC:
 54C30 Real-valued functions in general topology 49J22 Optimal control problems with integral equations (existence) (MSC2000) 54C60 Set-valued maps in general topology 54C65 Selections in general topology 46E40 Spaces of vector- and operator-valued functions
Full Text:
##### References:
  Arkin, V. I.: On an infinite dimensional analog of problems of non-convex programming,Kibernetika No. 2 (1967), 87-93 (in Russian).  Arkin, V. I. and Levin, V. L.: Convexity of ranges of convex integrals, measurable choice theorems and variational problems,Uspekhi Mat. Nauk 27(3) (1972), 21-77 (in Russian); English translation:Russian Math. Surveys 27(3) (1972), 21-85. · Zbl 0256.49025  Castaing, C.: Sur les multi-applications mesurables, Thèse, Caen, 1967; partly published inRevue française d’informatique et de recherche operationelle 1 (1967), 91-126.  Castaing, C. and Valadier, M.:Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, Heidelberg, New York, 1977. · Zbl 0346.46038  Dvoretzky, A., Wald, A., and Wolfowitz, J.: Relations among certain ranges of vector measures,Pacific J. Math. 1 (1951), 59-74. · Zbl 0044.15002  Grothendieck, A.:Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc.16 (1955). · Zbl 0123.30301  Himmelberg, C. J.: Measurable relations,Fund. Math. 87 (1975), 53-72. · Zbl 0296.28003  Kucia, A. and Nowak, A.: Relations among some classes of functions in mathematical programming, in E. Vilkas and R. Tamasiunas (eds),Decision Making in Economic Models, Mathematical Methods in Social Sciences, 22, Vilnius, 1989, pp. 29-33. · Zbl 0742.49009  Levin, V. L.: Measurable sections of set-valued mappings and projections of measurable sets,Funktsional Anal. i Prilozhen. 12(2) (1978), 40-45 (in Russian); English translation:Funct. Anal. Appl. 12 (1978).  Levin, V. L.: On Borel sections of set-valued mappings,Sibirsk. Mat. J. 19(3) (1978), 617-623 (in Russian); English translation:Siberian Math. J. (1978). · Zbl 0398.54027  Levin, V. L.: Measurable selections of multivalued mappings into topological spaces and upper envelopes of Carathéodory integrands,Doklady Akad. Nauk USSR 252(3) (1980), 535-539 (in Russian); English translation:Soviet Math. Dokl. 21(3) (1980), 771-775.  Levin, V. L.:Convex Analysis in Spaces of Measurable Functions and Its Applications in Mathematics and Economics, Nauka, Moscow, 1985 (in Russian). · Zbl 0617.46035  Levin, V. L.: Measurable selections of multifunctions with bianalytic graphs and ?-compact values,Trudy Moskov. Mat. Obshck. 54 (1992), 3-28 (in Russian); English translation:Trans. Moscow Math. Soc. (1993), 1-22. · Zbl 0771.28010  Lyapunov, A. A.: On completely additive vector functions,Izv. Akad. Nauk, ser. Math. 4(6) (1940), 465-478 (in Russian).  Neustadt, L.: The existence of optimal controls in the absence of convexity conditions,J. Math. Anal. Appl. 7 (1964), 110-118. · Zbl 0115.13304 · doi:10.1016/0022-247X(63)90081-7  Novikov, P. S.: On projections of certain B-sets,Dokl. Akad. Nauk USSR 23(9) (1939), 863-864 (in Russian). · Zbl 0061.09505  Rockafellar, R. T.: Convex integral functionals and duality, in E. Zarantonello (ed.),Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 215-236. · Zbl 0295.49006  Rockafellar, R. T.: Integral functionals, normal integrands and measurable selections, in G. P. Gossezet al. (eds),Proc. Colloq. Nonlinear Operators and the Calculus of Variations, Lecture Notes in Math. 543, Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp. 157-207. · Zbl 0374.49001  Wagner, D. H.: Survey of measurable selection theorems,SIAM J. Control Optim. 15(5) (1977), 859-903. · Zbl 0407.28006 · doi:10.1137/0315056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.