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

Kucia, Anna; Nowak, Andrzej

Tatra Mt. Math. Publ. 14, 261-267 (1998).

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.

Reviewer: Petr Holický (Praha))

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

Keywords:

measurable selections; Marczewski function; Carathéodory function; measurable compact-valued multifunctions

Citations:

Zbl 0351.90069; Zbl 0316.90080; Zbl 0767.28010; Zbl 0202.18401