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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)
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: DOI
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