Solution of the Baire order problem of Mauldin

Balcerzak, Marek; Rogowska, Dorota

doi:10.1090/S0002-9939-1995-1283538-5

Solution of the Baire order problem of Mauldin
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by Marek Balcerzak and Dorota Rogowska

Proc. Amer. Math. Soc. 123 (1995), 3413-3416

DOI: https://doi.org/10.1090/S0002-9939-1995-1283538-5

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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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Sur les fonctions représentables analytiquement et les ensembles de première catégorie

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Sur les fonctions représentable analytiquement

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