## Optimal measurable selections via Marczewski function.(English)Zbl 0945.28008

The authors give several remarks concerning the existence of measurable selections of measurable compact-valued multifunctions that maximize some Carathéodory-type control functions. It is shown that a theorem given by M. Schäl [Arch. Math. 25, 219-224 (1974; Zbl 0351.90069) and Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 179-196 (1975; Zbl 0316.90080)], and that is proved also by J. Burgess and A. Maitra [Proc. Am. Math. Soc. 116, No. 4, 1101-1106 (1992; Zbl 0767.28010)], can be proved by a reduction to a theorem of K. Hinderer [“Foundations of non-stationary dynamic programming with discrete time parameter” (1970; Zbl 0202.18401)]. The last mentioned theorem concerns the same problem with an upper semi-continuous compact-valued multifunction and an upper semi-continuous control function. The Marczewski embedding into the Cantor set and former results of the authors on the Carathéodory function are used.

### MSC:

 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 90C99 Mathematical programming 54C60 Set-valued maps in general topology

### Citations:

Zbl 0351.90069; Zbl 0316.90080; Zbl 0767.28010; Zbl 0202.18401