Effective refining of Borel coverings
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- by Gabriel Debs and Jean Saint Raymond PDF
- Trans. Amer. Math. Soc. 361 (2009), 2831-2869 Request permission
Abstract:
Given a countable family $(\mathbf {\Gamma }_i)_{i\in I}$ of additive or multiplicative Baire classes ($\mathbf {\Gamma }_i=\mathbf {\Sigma }^0_{\xi _i}$ or $\mathbf {\Pi }^0_{\xi _i}$) we investigate the following complexity problem: Let $(A_i)_{i\in I}$ be a Borel covering of $\omega ^\omega$ and assume that there exists some covering $(B_i)_{i\in I}$ with $B_i\subset A_i$ and $B_i\in \mathbf {\Gamma }_i$ for all $i$; can one find such a family $(B_i)_{i\in I}$ in $\varDelta ^1_1(\alpha )$ where $\alpha \in \omega ^\omega$ is any reasonable code for the families $(A_i)_{i\in I}$ and $(\mathbf {\Gamma }_i)_{i\in I}$? The main result of the paper will give a full characterization of those families $(\mathbf {\Gamma }_i)_{i\in I}$ for which the answer is positive. For example we will show that this is the case if $I$ is finite or if all the Baire classes $\mathbf {\Gamma }_i$ are additive, but in the general case the answer depends on the distribution of the multiplicative Baire classes inside the family $(\mathbf {\Gamma }_i)_{i\in I}$.References
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Additional Information
- Gabriel Debs
- Affiliation: Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4 place Jussieu, F-75252 Paris Cedex 05, France
- MR Author ID: 55795
- Email: debs@math.jussieu.fr
- Jean Saint Raymond
- Affiliation: Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4 place Jussieu, F-75252 Paris Cedex 05, France
- Email: raymond@math.jussieu.fr
- Received by editor(s): January 30, 2006
- Published electronically: January 22, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2831-2869
- MSC (2000): Primary 03E15; Secondary 03E45, 54H05
- DOI: https://doi.org/10.1090/S0002-9947-09-04930-7
- MathSciNet review: 2485410