Additive Families of Low Borel Classes and Borel Measurable Selectors

Spurný, J. ; Zelený, M.

Canadian mathematical bulletin, Tome 54 (2011) no. 1, pp. 180-192

Résumé

An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a $\sigma $ -discrete refinement. We confirm the conjecture for point-countable $\Pi _{3}^{0}$ -additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol.

Détail

DOI : 10.4153/CMB-2010-088-8

Mots-clés : 54H05, 54E35, σ-discrete refinement, Borel-additive family, measurable selection

Bibliographie

[1] [1] Engelking, R., General topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Google Scholar

[2] [2] Fleissner, W. G., An axiom for nonseparable Borel theory. Trans. Amer. Math. Soc. 251(1979), 309–328. doi:10.2307/1998696 Google Scholar

[3] [3] Fremlin, D. H., Measure-additive coverings and measurable selectors. Dissertationes Math. 260(1987), 116 pp. Google Scholar

[4] [4] Hansell, R. W., Borel measurable mappings for nonseparable metric spaces. Trans. Amer. Math. Soc. 161(1971), 145–169. doi:10.2307/1995934 Google Scholar

[5] [5] Hansell, R. W., On characterizing non-separable analytic and extended Borel sets as types of continuous images. Proc. London. Math. Soc. 28(1974), 683–699. doi:10.1112/plms/s3-28.4.683 Google Scholar

[6] [6] Hansell, R. W., Point-countable Souslin-additive families and σ–discrete reductions. In: General topology and its relation to modern analysis and algebra. V. (Prague, 1981), Sigma Series in Pure Mathematics, 3, Heldermann, Berlin, 1983, pp. 254–260. Google Scholar

[7] [7] Hansell, R. W., Hereditarily additive families in descriptive set theory and Borel measurable multimaps. Trans. Amer. Math. Soc. 278(1983), no. 2, 725–749. doi:10.2307/1999181 Google Scholar

[8] [8] Hansell, R. W., A measurable selection and representation theorem in nonseparable spaces. In: Measure theory, Oberwolfach 1983, Lecture Notes in Math., 1089, Springer, Berlin, 1984, pp. 86–94. Google Scholar

[9] [9] Hansell, R. W., F-set covers of analytic spaces and first class selectors. Proc. Amer. Math. Soc. 96(1986), no. 2, 365–371. doi:10.2307/2046182 Google Scholar

[10] [10] Hansell, R. W., Nonseparable analytic metric spaces and quotient maps. Topology Appl. 85(1998), no. 1–3, 143–152. doi:10.1016/S0166-8641(97)00146-6 Google Scholar

[11] [11] Kaniewski, J. and Pol, R., Borel-measurable selectors for compact-valued mappings in the non-separable case. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23(1975), no. 10, 1043–1050. Google Scholar

[12] [12] Kechris, A. S., Classical descriptive set theory. Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. Google Scholar

[13] [13] Pol, R., Note on decompositions of metrizable spaces. II. Fund. Math. 100(1978), no. 2, 129–143. Google Scholar

[14] [14] Rogers, C. E. and Jayne, J. E., K-analytic sets. In: Analytic sets, Academic Press, London, 1980, pp. 1–181. Google Scholar

[15] [15] Spurný, J., F-additive families and the invariance of Borel classes. Proc. Amer. Math. Soc. 133(2005), no. 3, 905–915. doi:10.1090/S0002-9939-04-07587-2 Google Scholar

[16] [16] Spurný, J., G-additive families in absolute Souslin spaces and Borel measurable selectors. Topology Appl. 154(2007), no. 15, 2779–2785. doi:10.1016/j.topol.2007.05.012 Google Scholar

[17] [17] Stone, A. H., Nonseparable Borel sets. II. General Topology and Appl. 2(1972), 249–270. doi:10.1016/0016-660X(72)90010-4 Google Scholar

[18] [18] Stone, A. H., Analytic sets in non-separable metric spaces. In: Analytic sets, Academic Press, London, 1980, pp. 471–480. Google Scholar