zbMATH — the first resource for mathematics

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)

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
90C39 Dynamic programming
Full Text: DOI
[1] Lester E. Dubins and Leonard J. Savage, Inequalities for stochastic processes (how to gamble if you must), Dover Publications, Inc., New York, 1976. Corrected republication of the 1965 edition. · Zbl 0359.60002
[2] C. J. Himmelberg, T. Parthasarathy, and F. S. VanVleck, Optimal plans for dynamic programming problems, Math. Oper. Res. 1 (1976), no. 4, 390 – 394. · Zbl 0368.90134 · doi:10.1287/moor.1.4.390 · doi.org
[3] K. Kuratowski and A. Mostowaski, Set theory, 2nd rev. ed., North-Holland, Amsterdam, 1976.
[4] Ashok Maitra, Discounted dynamic programming on compact metric spaces, Sankhyā Ser. A 30 (1968), 211 – 216. · Zbl 0187.17702
[5] A. Maitra and B. V. Rao, Generalizations of Castaing’s theorem on selectors, Colloq. Math. 44 (1981), 99-104. · Zbl 0427.28010
[6] A. S. Nowak and T. E. S. Raghavan, Existence of stationary correlated equilibria with symmetric information for discounted stochastic games, Math. Oper. Res. 17 (1992), no. 3, 519 – 526. · Zbl 0761.90112 · doi:10.1287/moor.17.3.519 · doi.org
[7] Ulrich Rieder, Measurable selection theorems for optimization problems, Manuscripta Math. 24 (1978), no. 1, 115 – 131. · Zbl 0385.28005 · doi:10.1007/BF01168566 · doi.org
[8] Manfred Schäl, A selection theorem for optimization problems, Arch. Math. (Basel) 25 (1974), 219 – 224. · Zbl 0351.90069 · doi:10.1007/BF01238668 · doi.org
[9] Manfred Schäl, Conditions for optimality in dynamic programming and for the limit of \?-stage optimal policies to be optimal, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), no. 3, 179 – 196. · Zbl 0316.90080 · doi:10.1007/BF00532612 · doi.org
[10] Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1 – 56. · Zbl 0207.00905 · doi:10.2307/1970696 · doi.org
[11] Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optimization 15 (1977), no. 5, 859 – 903. · Zbl 0407.28006 · doi:10.1137/0315056 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.