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Analytic ideals and their applications. (English) Zbl 0932.03060

Summary: We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal \(I\), either the ideal \(\{X\subset \omega\times \omega:\exists n X\subset \{0,1,\dots, n\}\times \omega\}\) is Rudin-Keisler below \(I\), or \(I\) is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We study this class of ideals and characterize, for example, when the ideals in it are \(F_\sigma\) or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of \(\sigma\)-ideals of \(\mu\)-zero sets for Maharam submeasures \(\mu\) on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author.

MSC:

03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
22A05 Structure of general topological groups
28A12 Contents, measures, outer measures, capacities
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[1] Balcerzak, M.; Roslanowski, A., On Mycielski ideals, (Proc. Amer. Math. Soc., 110 (1990)), 243-250 · Zbl 0708.04002
[2] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis, I (1963), Springer: Springer New York · Zbl 0115.10603
[3] Kechris, A. S., Countable sections for locally compact group actions, Ergodic Theory Dyn. Systems, 12, 283-295 (1992) · Zbl 0761.28014
[4] Kechris, A. S., Classical Descriptive Set Theory (1995), Springer: Springer Berlin · Zbl 0819.04002
[5] Kechris, A. S.; Louveau, A., A classification of hypersmooth equivalence relations, J. Amer. Math. Soc., 10, 215-242 (1997) · Zbl 0865.03039
[6] Kechris, A. S.; Louveau, A.; Woodin, H., The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc., 301, 263-288 (1987) · Zbl 0633.03043
[7] Kechris, A. S.; Solecki, S., Approximating analytic by Borel sets and definable chain conditions, Israel J. Math., 89, 343-356 (1995) · Zbl 0827.54023
[8] Kunen, K., Random and Cohen reals, (Handbook of Set-Theoretic Topology (1984), Elsevier: Elsevier Amsterdam), 887-911
[9] Louveau, A.; Velickovic, B., A note on Borel equivalence relations, (Proc. Amer. Math. Soc., 120 (1994)), 255-259 · Zbl 0794.04002
[10] Mathias, A. R.D., A remark on rare filters, (Hajnal, A.; etal., Infinite and Finite Sets. Infinite and Finite Sets, Colloq. Math. Soc. Janos Bolyai, 10 (1975), North-Holland: North-Holland Amsterdam) · Zbl 0342.02050
[11] K. Mazur, A modification of Louveau and Velickovic construction for \(F_σ\); K. Mazur, A modification of Louveau and Velickovic construction for \(F_σ\) · Zbl 0936.03046
[12] Mycielski, J., Some new ideals of sets on the real line, (Colloq. Math., 20 (1969)), 71-76 · Zbl 0203.05701
[13] J. Roslanowski, S. Shelah, Norms on possibilities II: more \(ccc^ω\); J. Roslanowski, S. Shelah, Norms on possibilities II: more \(ccc^ω\) · Zbl 0889.03036
[14] Solecki, S., Analytic ideals, Bull. Symb. Logic, 2, 339-348 (1996) · Zbl 0862.04002
[15] Talagrand, M., Compacts de fonctions mesurables et filtres non mesurables, Studia Math., 67, 13-43 (1980) · Zbl 0435.46023
[16] Todorcevic, S., Analytic gaps, Fund. Math., 150, 55-66 (1996) · Zbl 0851.04002
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