Point-finite Borel-additive families are of bounded class

Author:
R. W. Hansell

Journal:
Proc. Amer. Math. Soc. **83** (1981), 375-378

MSC:
Primary 54H05; Secondary 04A15

DOI:
https://doi.org/10.1090/S0002-9939-1981-0624935-8

MathSciNet review:
624935

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following theorem, which answers a question originally raised by J. Kaniewski and R. Pol:

Theorem. *If* *is a point-finite family of subsets of a metric space* *such that the union of every subfamily is a Borel set of* , *then there exists a fixed countable ordinal* *such that each member of* *is a Borel set of class* *in* .

The proof is given in the general setting of abstract measurable spaces. An application is made to the study of measurable selectors for compact-valued mappings and to the Borel measurability of graphs.

**[1]**W. G. Fleissner,*An axiom for nonseparable Borel theory*, Trans. Amer. Math. Soc.**251**(1979), 309-328. MR**531982 (83c:03043)****[2]**W. G. Fleissner, R. W. Hansell and H. J. K. Junnila,*PMEA implies Proposition*, General Topology Appl. (to appear). MR**651508 (83f:54042)****[3]**R. W. Hansell,*Borel-additive families and Borel maps in metric spaces*, General Topology and Modern Analysis, edited by L. F. McAuley and M. M. Rao, Academic Press, New York, 1981. MR**619067 (82i:54069)****[4]**C. A. Rogers (Editor),*Analytic sets*, Academic Press, New York, 1980. MR**608794 (82m:03063)****[5]**J. Kaniewski and R. Pol,*Borel-measurable selectors for compact-valued mappings in the nonseparable case*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**23**(1975), 1043-1050. MR**0410657 (53:14405)****[6]**K. Kuratowski,*Topology*, Vol. 1, Academic Press, New York; PWN, Warsaw, 1966. MR**0217751 (36:840)****[7]**D. Preiss,*Completely additive disjoint systems of Baire sets is of bounded class*, Comment. Math. Univ. Carolinae**15**(1972), 341-344. MR**0346116 (49:10842)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0624935-8

Keywords:
Borel-additive family,
point-finite family,
Borel classifications,
measurable selectors

Article copyright:
© Copyright 1981
American Mathematical Society