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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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##### References:
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