Borel measurable mappings for nonseparable metric spaces

Abstract:

The main object of this paper is the extension of part of the basic theory of Borel measurable mappings, from the “classical” separable metric case, to general metric spaces. Although certain results of the standard theory are known to fail in the absence of separability, we show that they continue to hold for the class of “$\sigma$-discrete” mappings. This class is shown to be quite extensive, containing the continuous mappings, all mappings with a separable range, and any Borel measurable mappings whose domain is a Borel subset of a complete metric space. The last result is a consequence of our Basic Theorem which gives a topological characterization of those collections which are the inverse image of an open discrete collection under a Borel measurable mapping. Such collections are shown to possess a strong type of $\sigma$-discrete refinement. The properties of $\sigma$-discrete mappings together with the known properties of “locally Borel” sets allow us to extend, to general metric spaces, well-known techniques used for separable spaces. The basic properties of “complex” and “product” mappings, well known for separable spaces, are proved for general metric spaces for the class of $\sigma$-discrete mappings. A consequence of these is a strengthening of the basic theorem of the structure theory of nonseparable Borel sets due to A. H. Stone. Finally, the classical continuity properties of Borel measurable mappings are extended, and, in particular, a generalization of the famous theorem of Baire on the points of discontinuity of a mapping of class 1 is obtained.