The Baire order of the functions continuous almost everywhere
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- by R. D. Mauldin
- Proc. Amer. Math. Soc. 41 (1973), 535-540
- DOI: https://doi.org/10.1090/S0002-9939-1973-0323966-0
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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.References
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Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 535-540
- MSC: Primary 26A21
- DOI: https://doi.org/10.1090/S0002-9939-1973-0323966-0
- MathSciNet review: 0323966