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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The Baire order of the functions continuous almost everywhere


Author: R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 51 (1975), 371-377
MSC: Primary 26A21
DOI: https://doi.org/10.1090/S0002-9939-1975-0372128-1
MathSciNet review: 0372128
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Abstract: Let $ S$ be a complete and separable metric space and $ \mu $ a $ \sigma $-finite, complete Borel measure on $ S$ with $ \mu (S) > 0$. Let $ \Phi $ be the family of all real-valued functions defined on $ S$ whose set of points of discontinuity is of $ \mu $-measure 0. Let $ {B_\alpha }(\Phi )$ be the functions of Baire's class $ \alpha $ generated by $ \Phi $. It is shown that $ {B_1}(\Phi ) = {B_2}(\Phi )$ if and only if $ \mu $ is a purely atomic measure whose set of atoms forms a scattered subset of $ S$ and that if $ {B_1}(\Phi ) \ne {B_2}(\Phi )$, then the Baire order of $ \Phi $ is $ {\omega _1}$; in other words, if $ 0 \leq \alpha < {\omega _1}$, then $ {B_\alpha }(\Phi ) \ne {B_{\alpha + 1}}(\Phi )$. This answers a generalized version of a problem raised by Sierpinski and Felsztyn. An example is given of a normal space with Borel order 2 and Baire order $ {\omega _1}$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0372128-1
Keywords: Borel measure, Baire's class $ \alpha $, 0-dimension, scattered
Article copyright: © Copyright 1975 American Mathematical Society