an:00026151
Zbl 0742.49009
Kucia, A.; Nowak, A.
Relations among some classes of functions in mathematical programming
EN
Mat. Metody Sots. Naukakh 22, 29-33 (1989).
00182167
1989
j
49J45 54C30 26B99 49N99 90C99
Carath??odory functions; normal integrands
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 Carath??odory 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. \textit{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 \(F_ 2\neq F_ 3\).
A.Kucia
Zbl 0374.49001