Measurability of multifunctions of two variables

Volume 452 / 2008

Gra/zyna Kwieci/nska Dissertationes Mathematicae 452 (2008), 1-67 MSC: 54C60, 28B20, 46G05, 26E25, 28A35, 28A15, 58C20. DOI: 10.4064/dm452-0-1


\def\bull{\setbox0=\hbox{\hskip15pt$\bullet$} \noindent\kern\wd0\llap{$\bullet$} \ignorespaces } We consider multifunctions $F$ of two variables, with values in a topological space, whose first argument ranges over a measurable space, and second over a space with various possible structures: topological, metric (with a differentiation basis if needed), normed linear, etc. We are mainly interested in product measurability and superpositional measurability of $F$. Some connections between classes of multifunctions with these properties are considered. In Chapter 2, several product measurability results are proved for multifunctions which are measurable in the first and satisfy some special hypothesis in the second variable, e.g. \vskip4pt \bull are either right continuous or left continuous in some sense, \bull are approximately $h$-equicontinuous with respect to a differentiation basis, \bull are lower semicontinuous and upper quasi-continuous, \bull are both upper and lower strong quasi-continuous with respect to a differentiation basis, \bull are derivatives. \vskip4pt Chapter 3 is devoted to superpositional measurability of multifunctions. Some of the results of this chapter are consequences of results of Chapter 2 and Zygmunt's theorem on superpositional measurability of multifunctions which are measurable with respect to the product of a complete $\sigma$-field and the $\sigma$-field of Borel subsets of a Polish space. In general, a product measurable multifunction need not be superpositionally measurable. We prove that (in suitable spaces) multifunctions which are product measurable with respect to a $\sigma$-field more general than the product $\sigma$-field above and which also fulfill certain density conditions in the second variable are also superpositionally measurable. Counterexamples are also given to emphasize the need for some of the hypotheses.


  • Gra/zyna Kwieci/nskaInstitute of Mathematics
    University of Gda/nsk
    Wita Stwosza 57
    80-952 Gda/nsk, Poland

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