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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

##### MSC:
 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
##### Keywords:
Carathéodory functions; normal integrands