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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On Borel measures and Baire's class $ 3$


Author: R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 39 (1973), 308-312
MSC: Primary 26A21
MathSciNet review: 0316640
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Abstract: Let $ S$ be a complete and separable metric space and $ \mu $ a $ \sigma $-finite, complete Borel measure on $ S$. Let $ \Phi $ be the family of all real-valued functions, continuous $ \mu $-a.e. Let $ {B_\alpha }(\Phi )$ be the functions of Baire's class $ \alpha $ generated by $ \Phi $. It is shown that if $ \mu $ is not a purely atomic measure whose set of atoms form a dispersed subset of $ S$, then $ {B_2}(\Phi ) \ne {B_{{\omega _1}}}(\Phi )$, where $ {\omega _1}$ denotes the first uncountable ordinal.


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  • [1] F. Hausdorff, Mengenlehre, 3rd ed., de Gruyter, Berlin, 1937; English transl., 2nd ed., Chelsea, New York, 1962. MR 25 #4999.
  • [2] R. Daniel Mauldin, On the Baire system generated by a linear lattice of functions, Fund. Math. 68 (1970), 51–59. MR 0273363 (42 #8243)
  • [3] R. Daniel Mauldin, Some examples of 𝜎-ideals and related Baire systems, Fund. Math. 71 (1971), no. 2, 179–184. MR 0293028 (45 #2108)
  • [4] R. Daniel Mauldin, 𝜎-ideals and related Baire systems, Fund. Math. 71 (1971), no. 2, 171–177. MR 0293027 (45 #2107)
  • [5] John E. Jayne, Space of Baire functions, Baire classes, and Suslin sets, Dissertation, Columbia University, New York, 1971.
  • [6] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York, 1966. MR 0217751 (36 #840)
  • [7] Z. Semadeni, Spaces of continuous functions. II, Studia Math. 16 (1957), 193–199. MR 0092942 (19,1184c)
  • [8] S. Banach, Über analytisch darstellbare Operationen in abstrakten Raum, Fund. Math. 17 (1931), 283-295.
  • [9] R. Baire, Sur la représentation des fonctions discontinues, Acta. Math. 30 (1905); ibid. 32 (1909).
  • [10] L. Kantorovitch, Sur les suites des fonctions presque partout continues, Fund. Math. 16 (1930), 25-28.
  • [11] N. Luzin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris, 1930.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0316640-8
PII: S 0002-9939(1973)0316640-8
Keywords: Borel measure, Baire function, Baire class $ \alpha $, dispersed set, ambiguous sets
Article copyright: © Copyright 1973 American Mathematical Society