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Proceedings of the American Mathematical Society
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. D. Mauldin
Journal: Proc. Amer. Math. Soc. 41 (1973), 535-540
MSC: Primary 26A21
MathSciNet review: 0323966
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Abstract: Let $ \Phi $ be the family of all real-valued functions defined on the unit interval $ I$ which are continuous except for a set of Lebesgue measure zero. Let $ {\Phi _0}$ be $ \Phi $ and for each ordinal $ \alpha $, let $ {\Phi _\alpha }$ be the family of all pointwise limits of sequences taken from $ \bigcup\nolimits_{\gamma < \alpha } {{\Phi _\gamma }} $ Then $ {\Phi _{{\omega _1}}}$ is the Baire family generated by $ \Phi $. It is proven here that if $ 0 < \alpha < {\omega _1}$, then $ {\Phi _\alpha } \ne {\Phi _{{\omega _1}}}$. The proof is based upon the construction of a Borel measurable function $ h$ from $ I$ onto the Hilbert cube $ Q$ such that if $ x$ is in $ Q$, then $ {h^{ - 1}}(x)$ is not a subset of an $ {F_\sigma }$ set of Lebesgue measure zero.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0323966-0
PII: S 0002-9939(1973)0323966-0
Keywords: Lebesgue measure zero, Baire class $ \alpha $, universal function, Hilbert cube
Article copyright: © Copyright 1973 American Mathematical Society