Borel structures on the set of Borel mappings

Abstract

Let Y and Z be two Borel spaces. By B(Y,Z) we denote the set of all Borel maps of Y into Z. In Aumann (1961) [2] and Rao (1971) [10] the authors tried to generalize the results of R. Arens and J. Dugundji (see Arens and Dugundji (1951) [1]) for Borel spaces. Unfortunately as R.J. Aumann observed in Aumann (1961) [2], the results of Arens and Dugundji (1951) [1] are not true for Borel spaces, since for some of the simplest Borel spaces for example it is impossible to defined a Borel structure on the set B(Y,Z) such that the map e:B(Y,Z)×Y→Z with e(f,y)=f(y) for every f∈B(Y,Z) and y∈Y is Borel. Even if we consider the discrete structure on B(Y,Z), then e will not in general be Borel. It is for this reason that in Aumann (1961) [2] and Rao (1971) [10] the authors studied subsets F of B(Y,Z) and Borel structures on F such that the restriction of the map e on F×Y is Borel.In this paper we study the above problem and try to generalize the results presented in Arens and Dugundji (1951) [1] for Borel spaces. More precisely in Section 1 the paper preliminaries are given. In Sections 2 and 3 we give and study Borel A-splitting and A-admissible structures on B(Y,Z), where A is an arbitrary family of Borel spaces, and prove that there exists at most one Borel structure on B(Y,Z) which is both Borel splitting and admissible. When this structure exists, it coincides with the greatest Borel splitting structure, which always exists. We also present and study some special Borel structures on B(Y,Z). In Section 4 some remarks for Borel structures on B(Y,Z) are stated. In Section 5 we define and study some relations between the Borel structures of the set B(Y,Z) and the Borel structures of the set BZ(Y) consisting of all subsets f−1(B) of Y, where f∈B(Y,Z) and B is an element of the Borel structure of Z, concerning the notions of Borel A-splitting and Borel A-admissible Borel structures. Finally, some open questions for Borel structures on the set of Borel mappings are posed.