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Nonexistence of measurable optimal selections. (English) Zbl 0767.28010
Let \((T,{\mathcal T})\) be a measurable space, \(Y\) a separable metric space, \(G\) a measurable and compact-valued multifunction from \(T\) to \(Y\), and \(v\) a real-valued function on the graph of \(G\). M. Schäl [Arch. Math. 25, 219-224 (1974; Zbl 0351.90069)] proved that if \(v\) is the pointwise limit of a decreasing sequence of functions which are product-measurable and continuous in \(y\), then there exists an optimal measurable selection, i.e., a measurable selector \(g\) of \(G\) such that \(v(t,g(t))=\sup\{v(t,y):y\in G(t)\}\), \(t\in T\). The authors give another proof of this theorem, but the main result of the paper is a counterexample to the conjecture that assumptions on \(v\) can be replaced by the weaker condition: \(v\) is product-measurable and upper semicontinuous in \(y\). The key role plays the construction of a function \(f\) from a separable metric space \(X\) into a compact metric space \(Y\) such that the graph of \(f\) is a Borel subset of \(X\times Y\), but \(f\) is not Borel measurable. It is based on a coding of Borel subsets of the real line, due to R. M. Solovay [Ann. Math., II. Ser. 92, 1-56 (1970; Zbl 0207.009)].
Reviewer: A.Nowak (Katowice)

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
90C39 Dynamic programming
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