# zbMATH — the first resource for mathematics

Carathéodory selections and the Scorza Dragoni property. (English) Zbl 0649.28011
The authors state first the following Scorza Dragoni type result for multifunctions. Let T be a separable metric and complete space and let $$\mu$$ be a finite measure on T. Let X be a separable complete metric space equipped with its Borel structure. Let F be a multifunction from $$T\times X$$ taking values in X such that F is measurable and such that $$F(t,\cdot)$$ is lower semicontinuous on X for each fixed $$t\in T.$$ Then, for every $$\epsilon >0,$$ there exists a compact set $$K\subset T$$ with $$\mu (T\setminus K)\leq \epsilon$$ and such that $$F|_{K\times X}$$ is lower semicontinuous.
Application to the existence of Carathéodory selection is given and the authors show that the condition of measurability with respect to the variables $$(t,x)$$ cannot be dropped.
Note of the reviewer: The main result given by the authors is a consequence of a more general result published by the reviewer in Trav. Semin. d’Anal. Convexe, Vol. 6, Montpellier 1976, Exposé No.6 (1976; Zbl 0356.46045) and the approach to reduce the problem of finding a Carathéodory selection to the problem of finding continuous selection for a lower semi continuous mapping is also exploited in this previous paper by the reviewer.
Reviewer: Ch.Castaing

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
Zbl 0356.46045
Full Text:
##### References:
 [1] Castaing, C, Une nouvelle extension du théorème de dragoni-scorza, C. R. acad. sci. Paris ser. A, 271, 396-398, (1970) · Zbl 0199.49302 [2] Castaing, C, Sur l’existence des sections separement mesurables et separement continues d’une multi-application, (), Ex. 14 · Zbl 0353.46031 [3] Castaing, C; Valadier, M, Convex analysis and measurable multifunctions, () · Zbl 0346.46038 [4] Curtis, D.W, Applications of a selection theorem to hyperspace contractibility, Canad. J. math., 37, 747-759, (1985) · Zbl 0563.54008 [5] Fryszkowski, A, Caratheodory type selections of set-valued maps of two variables, Bull. acad. polon. sci. ser. sci. math. astronom. phys., 25, 41-46, (1977) · Zbl 0358.54002 [6] Fryszkowski, A, Continuous selections for a class of non-convex multivalued maps, Studia math., 76, 163-174, (1983) · Zbl 0534.28003 [7] Himmelberg, C.F; Van Vleck, F.S, An extension of Brunovsky’s scorza draconi type theorem for unbounded set-valued functions, Math. slovaca, 26, 47-52, (1976) · Zbl 0328.28004 [8] Itoh, S, Random fixed point theorems with applications to random differential equations in Banach spaces, J. math. anal. appl., 62, 261-273, (1969) · Zbl 0407.60069 [9] Jacobs, M.I, Measurable multivalued mappings and Lusin’s theorem, Trans. amer. math. soc., 134, 471-481, (1968) · Zbl 0169.06801 [10] Kim, T; Prikry, K; Yannelis, N.C, Caratheodory-type selections and random fixed point theorems, J. math. anal. appl., 122, 393-407, (1987) · Zbl 0629.28007 [11] Kim, T; Prikry, K; Yannelis, N.C, Equilibria in abstract economies with a measure space of agents and with an infinite dimensional strategy space, (1985), University of Minnesota, preprint [12] Kuratowski, K, Topology I, (1966), Academic Press New York · Zbl 0158.40901 [13] Michael, E.A, Continuous selections I, Ann. of math., 63, 363-382, (1956) · Zbl 0071.15902 [14] Michael, E.A, A survey of continuous selections, (), 54-58 [15] Papageorgiou, N.S, On measurable multifunctions with applications to random multivalued equations, (1985), University of Illinois Urbana, preprint [16] Rybinski, L, On caratheodory type selections, Fund. math., 125, 187-193, (1985) · Zbl 0614.28005 [17] Dragoni, G.Scorza, Una theorema sulla funzioni continue rispetto ad una i misurable rispetto at ultra variable, (), 102-106 · Zbl 0032.19702 [18] Wagner, D.H, Survey of measurable selection theorems, SIAM J. control optim., 15, 859-903, (1977) · Zbl 0407.28006
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.