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Proceedings of the American Mathematical Society

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Solution of the Baire order problem of Mauldin


Authors: Marek Balcerzak and Dorota Rogowska
Journal: Proc. Amer. Math. Soc. 123 (1995), 3413-3416
MSC: Primary 54H05; Secondary 04A15, 26A21
MathSciNet review: 1283538
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Abstract: Let X be an uncountable Polish space, and let I be a proper $ \sigma $-ideal of subsets of X such that $ \{ x\} \in I$ for each $ x \in X$. Denote by $ {B_\alpha }(I),\alpha \leq {\omega _1}$, the Baire system generated by the family of functions $ f:X \to \mathbb{R}$ continuous I almost everywhere. We prove that if $ r(I) = \min \{ \alpha \leq {\omega _1}:{B_{\alpha + 1}}(I) = {B_\alpha }(I)\} $, then either $ r(I) = 1$ or $ r(I) = {\omega _1}$. This answers the problem raised by R. D. Mauldin in 1973.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1283538-5
Keywords: $ \sigma$-ideals, Baire classification, sets of the first category
Article copyright: © Copyright 1995 American Mathematical Society