Carathéodory selections and the Scorza Dragoni property.

*(English)*Zbl 0649.28011The authors state first the following Scorza Dragoni type result for multifunctions. Let T be a separable metric and complete space and let \(\mu\) be a finite measure on T. Let X be a separable complete metric space equipped with its Borel structure. Let F be a multifunction from \(T\times X\) taking values in X such that F is measurable and such that \(F(t,\cdot)\) is lower semicontinuous on X for each fixed \(t\in T.\) Then, for every \(\epsilon >0,\) there exists a compact set \(K\subset T\) with \(\mu (T\setminus K)\leq \epsilon\) and such that \(F|_{K\times X}\) is lower semicontinuous.

Application to the existence of Carathéodory selection is given and the authors show that the condition of measurability with respect to the variables \((t,x)\) cannot be dropped.

Note of the reviewer: The main result given by the authors is a consequence of a more general result published by the reviewer in Trav. Semin. d’Anal. Convexe, Vol. 6, Montpellier 1976, Exposé No.6 (1976; Zbl 0356.46045) and the approach to reduce the problem of finding a Carathéodory selection to the problem of finding continuous selection for a lower semi continuous mapping is also exploited in this previous paper by the reviewer.

Application to the existence of Carathéodory selection is given and the authors show that the condition of measurability with respect to the variables \((t,x)\) cannot be dropped.

Note of the reviewer: The main result given by the authors is a consequence of a more general result published by the reviewer in Trav. Semin. d’Anal. Convexe, Vol. 6, Montpellier 1976, Exposé No.6 (1976; Zbl 0356.46045) and the approach to reduce the problem of finding a Carathéodory selection to the problem of finding continuous selection for a lower semi continuous mapping is also exploited in this previous paper by the reviewer.

Reviewer: Ch.Castaing

##### MSC:

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

28C05 | Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures |

##### Keywords:

Scorza Dragoni type result for multifunctions; Carathéodory selection; continuous selection##### Citations:

Zbl 0356.46045
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\textit{Z. Artstein} and \textit{K. Prikry}, J. Math. Anal. Appl. 127, 540--547 (1987; Zbl 0649.28011)

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