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Relations among some classes of functions in mathematical programming. (English) Zbl 0742.49009
For a measurable space \((T,{\mathcal T})\) and a separable metric space \(X\) the following classes of functions \(f: T\times X\to [-\infty,+\infty)\) are considered: \(F_ 1=\{f:f\) is \({\mathcal T}\otimes {\mathcal B}(X)\)- measurable and upper semicontinuous in \(x\}\), \(F_ 2=\{f:f\) is a limit of a decreasing sequence of Carathéodory functions\(\}\), \(F_ 3=\{f:f\) is upper semicontinuous in \(x\), and the set-valued map \(t \to\{(x,r)\in X\times R:\;f(t,x)\geq r\}\) is measurable\(\}\). These families arise in optimization and mathematical economics. Elements of \(F_ 3\) are called normal integrands, cf. R. T. Rockafellar [in: Nonlin. Oper. Calc. Var., Summer Sch. Bruxelles 1975, Lect. Notes Math. 543, 157-207 (1976; Zbl 0374.49001)]. We study inclusions between these classes of functions; some of them were already known.
Always \(F_ 3\subset F_ 2\subset F_ 1\). If \(\mathcal T\) is closed under the Souslin operation and \(X\) is Souslin, then \(F_ 1=F_ 2=F_ 3\). If \(T\) and \(X\) are Souslin spaces and \({\mathcal T}={\mathcal B}(X)\), then \(F_ 1=F_ 2\). If \(X\) is \(\sigma\)-compact, then \(F_ 2=F_ 3\). We have examples that \(F_ 1\neq F_ 2\) and \(F_ 2\neq F_ 3\).
Reviewer: A.Kucia

49J45 Methods involving semicontinuity and convergence; relaxation
54C30 Real-valued functions in general topology
26B99 Functions of several variables
49N99 Miscellaneous topics in calculus of variations and optimal control
90C99 Mathematical programming