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Proceedings of the American Mathematical Society

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On Borel measures and Baire's class $ 3$


Author: R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 39 (1973), 308-312
MSC: Primary 26A21
DOI: https://doi.org/10.1090/S0002-9939-1973-0316640-8
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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DOI: https://doi.org/10.1090/S0002-9939-1973-0316640-8
Keywords: Borel measure, Baire function, Baire class $ \alpha $, dispersed set, ambiguous sets
Article copyright: © Copyright 1973 American Mathematical Society

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