Measurability of multifunctions of two variables

Grażyna Kwiecińska

GDML_Books, (2008), p.

Résumé

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. • are either right continuous or left continuous in some sense, • are approximately h-equicontinuous with respect to a differentiation basis, • are lower semicontinuous and upper quasi-continuous, • are both upper and lower strong quasi-continuous with respect to a differentiation basis, • are derivatives. 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 σ-field and the σ-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 σ-field more general than the product σ-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.

Détail

Zbl 1157.54007

EUDML-ID : urn:eudml:doc:285954