Applications of the Hausdorff approach to semicontinuous functions (Q2774500)

The authors consider a measurable space \((T,{\mathcal T})\), a metric space \(X\), a subset \(D\) of \(T\times X\) and the collection \(F(D)\) of all real-valued functions on \(D\) which are pointwise limits of increasing sequences of Carathéodory functions. A function \(f\:D\rightarrow {\mathbb R}\) is said to be Carathéodory if it is measurable with respect to the \(\sigma \)-field \({\mathcal D}=\{ D\cap A\:A\in {\mathcal T}\otimes {\mathcal B}(X)\}\), where \({\mathcal B}(X)\) is the Borel \(\sigma \)-field on \(X\). The class \(F(D)\) appears in optimization theory, e.g., in the study of optimal measurable selections [\textit{M. Schäl}, Arch. Mat. 25, 219-224 (1974; Zbl 0351.90069); Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 179-196 (1975; Zbl 0316.90080)]. NEWLINENEWLINENEWLINEThe main result of the paper (Theorem 3.2) is the following: Assume that \(X\) is separable, that each set \(D_{t}=\{x\in X\:(t,x)\in D\}\), \((t\in T)\), is nonvoid and that \(D\) has a Castaing representation \(U\), that is a countable family \(U\) of measurable functions \(u\:T\rightarrow X\) such that, for each \(t\in T\), the set \(U(t)=\{ u(t)\:u\in U\}\) is dense in \(D_{t}\). Assume also that, given a function \(f\:D\rightarrow {\mathbb R}\), we have: (a) for each \(u\in U\), the function \(t\rightarrow f(t,u(t))\) is measurable; (b) for each \(t\in T\) and each \(x\in D_{t}\), \(\liminf _{y\in U(t), y\rightarrow x} f(t,y)=f(t,x). \) Then \(f\in F(D)\) and, consequently, \(f\) is measurable. NEWLINENEWLINENEWLINEThe proof of this main result is based on the parametrization of a Hausdorff formula for lower semicontinuous functions. Various applications of this theorem are given and, to conclude the paper, the authors give some corollaries on the measurability of a multifunction of two variables.