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Convergence and approximation results for measurable multifunctions. (English) Zbl 0656.60010

The purpose of this note is to extend some of the results of G. Salinetti and R. Wets [Trans. Am. Math. Soc. 266, 275-289 (1981; Zbl 0501.28005)] on convergence and approximability of measurable multifunctions taking values in separable Banach spaces (Th. 3.1; Th. 3.2; Th. 3.3).
Dinh Quang Luu [Acta Math. Vietnam. 5, No.2, 141-143 (1980; Zbl 0496.28010)] gave the characterization of a.e. Hausdorff convergence of measurable multifunctions \((X_ n,X_{\infty})\) via the a.e. convergence of Castaing representations \((f^ i_ n,f^ i_{\infty})\) (compare with Salinetti-Wets’ results and with theorem 3.1 given by the authors). For a systematic study of the measurability of multifunctions defined on an abstract measurable space (\(\Omega\),\({\mathcal A})\) and taking values in a separable metric space, see Ch. Hess [Sémin. Anal. Convexe, Univ. Sci. Tech. Languedoc 15, Exp. No.9, 104 p. (1985; Zbl 0622.28010)].
Approximation, Mosco convergence, Borel structures on closed sets, measurability of inferior and superior limits etc. are investigated nicely.
Reviewer: Ch.Castaing

MSC:

60A99 Foundations of probability theory
54C40 Algebraic properties of function spaces in general topology
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