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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
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