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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Baire order of the functions continuous almost everywhere
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by R. Daniel Mauldin PDF
Proc. Amer. Math. Soc. 51 (1975), 371-377 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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