Borel measurable mappings for nonseparable metric spaces
HTML articles powered by AMS MathViewer
- by R. W. Hansell PDF
- Trans. Amer. Math. Soc. 161 (1971), 145-169 Request permission
Abstract:
The main object of this paper is the extension of part of the basic theory of Borel measurable mappings, from the âclassicalâ 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 â$\sigma$-discreteâ 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 âlocally Borelâ sets allow us to extend, to general metric spaces, well-known techniques used for separable spaces. The basic properties of âcomplexâ and âproductâ 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.References
-
S. Banach, Ăber analytisch darstellbare Operationen in abstrakten Räumen, Fund. Math. 17 (1931), 281-295.
- R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175â186. MR 43449, DOI 10.4153/cjm-1951-022-3
- A. G. Elâ˛kin, $A$-sets in complete metric spaces, Dokl. Akad. Nauk SSSR 175 (1967), 517â520 (Russian). MR 0214029 F. Hausdorff, Mengenlehre, de Gruyter, Berlin, 1937; English transl., Chelsea, New York, 1957. MR 19, 111.
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; PaĹstwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751 â, Quelques problèmes concernant les espaces mĂŠtriques non-sĂŠparables, Fund. Math. 25 (1935), 535. â, Sur le prolongement de lâhomĂŠomorphie, C. R. Acad. Sci. Paris 197 (1933), 1090. H. Lebesgue, J. Math. 1 (1905), 168. D. Montgomery, Non-separable metric spaces, Fund. Math. 25 (1935), 527-534.
- Kiiti Morita, Normal families and dimension theory for metric spaces, Math. Ann. 128 (1954), 350â362. MR 65906, DOI 10.1007/BF01360142 W. SierpiĹski, General topology, 2nd ed., Univ. of Toronto Press, Toronto, 1956.
- A. H. Stone, Non-separable Borel sets, Rozprawy Mat. 28 (1962), 41. MR 152457
- A. H. Stone, On $\sigma$-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655â666. MR 156789, DOI 10.2307/2373113
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 161 (1971), 145-169
- MSC: Primary 28.20; Secondary 54.00
- DOI: https://doi.org/10.1090/S0002-9947-1971-0288228-1
- MathSciNet review: 0288228