# Article

Full entry | PDF   (0.1 MB)
Keywords:
normal integrand; Carathéodory function
Summary:
Let $D\subset T\times X$, where $T$ is a measurable space, and $X$ a topological space. We study inclusions between three classes of extended real-valued functions on $D$ which are upper semicontinuous in $x$ and satisfy some measurability conditions.
References:
[1] Ash R.B.: Real Analysis and Probability. Academic Press, New York, 1972. MR 0435320
[2] Berliocchi H., Lasry J.-M.: Intégrandes normales et mesures paramétrées en calcul de variations. Bull. Soc. Math. France 101 (1973), 129-184. MR 0344980
[3] Burgess J., Maitra A.: Nonexistence of measurable optimal selections. Proc. Amer. Math. Soc. 116 (1992), 1101-1106. MR 1120505 | Zbl 0767.28010
[4] Christensen J.P.R.: Topology and Borel Structure. North Holland, Amsterdam, 1974. MR 0348724 | Zbl 0273.28001
[5] Himmelberg C.J.: Measurable relations. Fund. Math. 87 (1975), 53-72. MR 0367142 | Zbl 0296.28003
[6] Kucia A.: Some counterexamples for Carathéodory functions and multifunctions. submitted to Fund. Math.
[7] Kucia A., Nowak A.: On Baire approximations of normal integrands. Comment. Math. Univ. Carolinae 30:2 (1989), 373-376. MR 1014136 | Zbl 0685.28001
[8] Kucia A., Nowak A.: Relations among some classes of functions in mathematical programming. Mat. Metody Sots. Nauk 22 (1989), 29-33. MR 1111399 | Zbl 0742.49009
[9] Levin V.L.: Measurable selections of multivalued mappings into topological spaces and upper envelopes of Carathéodory integrands (in Russian). Dokl. Akad. Nauk SSSR 252 (1980), 535-539 English transl.: Sov. Math. Dokl. 21 (1980), 771-775. MR 0577834
[10] Levin V.L.: Convex Analysis in Spaces of Measurable Functions and its Applications to Mathematics and Economics (in Russian). Nauka, Moscow, 1985. MR 0809179
[11] Pappas G.S.: An approximation result for normal integrands and applications to relaxed controls theory. J. Math. Anal. Appl. 93 (1983), 132-141. MR 0699706 | Zbl 0521.49012
[12] Rockafellar R.T.: Integral functionals, normal integrands and measurable selections. in: Nonlinear Operators and Calculus of Variations (L. Waelbroeck, ed.), Lecture Notes in Mathematics 543, Springer, Berlin, 1976, pp. 157-207. MR 0512209 | Zbl 0374.49001
[13] Schäl M.: A selection theorem for optimization problem. Arch. Math. 25 (1974), 219-224. MR 0346632
[14] Wagner D.H.: Survey of measurable selection theorems. SIAM J. Control 15 (1977), 859-903. MR 0486391 | Zbl 0407.28006
[15] Zygmunt W.: Scorza-Dragoni property (in Polish). UMCS, Lublin, 1990.

Partner of