The structure of $\sigma$-ideals of compact sets
Authors:
A. S. Kechris, A. Louveau and W. H. Woodin
Journal:
Trans. Amer. Math. Soc. 301 (1987), 263-288
MSC:
Primary 03E15; Secondary 28A05, 42A63
DOI:
https://doi.org/10.1090/S0002-9947-1987-0879573-9
MathSciNet review:
879573
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Abstract: Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where $\sigma$-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of $\sigma$-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a $\sigma$-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that $\sigma$-ideals which are analytic or coanalytic must be actually either complete coanalytic or else ${G_\delta }$. In the second part we discuss (generators or as we call them here) bases for $\sigma$-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel $\sigma$-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of $\sigma$-ideals with other structural properties, like the countable chain condition, etc.
- N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Co., New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116
- John P. Burgess and R. Daniel Mauldin, Conditional distributions and orthogonal measures, Ann. Probab. 9 (1981), no. 5, 902–906. MR 628885
- D. Cenzer and R. D. Mauldin, Faithful extensions of analytic sets to Borel sets, Houston J. Math. 6 (1980), no. 1, 19–29. MR 575911
- Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760
- Jens Peter Reus Christensen, Necessary and sufficient conditons for the measurability of certain sets of closed subsets, Math. Ann. 200 (1973), 189–193. MR 334169, DOI https://doi.org/10.1007/BF01425230
- Claude Dellacherie, Ensembles analytiques, capacités, mesures de Hausdorff, Lecture Notes in Mathematics, Vol. 295, Springer-Verlag, Berlin-New York, 1972 (French). MR 0492152
- Claude Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin-New York, 1972 (French). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67. MR 0448504
- C. Dellacherie, D. Feyel, and G. Mokobodzki, Intégrales de capacités fortement sous-additives, Seminar on Probability, XVI, Lecture Notes in Math., vol. 920, Springer, Berlin-New York, 1982, pp. 8–40 (French). MR 658670 C. Dellacherie, Appendice á l’exposé précédent, ibid., pp. 29-40.
- Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel, Hermann, Paris, 1975 (French). Chapitres I à IV; Édition entièrement refondue; Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV; Actualités Scientifiques et Industrielles, No. 1372. MR 0488194
- Gérard Hillard, Une généralisation du théorème de Saint-Raymond sur les boréliens à coupes ${\cal K}_{\sigma }$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 16, A749–A751 (French, with English summary). MR 535803 W. Hurewicz, Relativ Perfecte Teile von Punktmengen und Mengen (A), Fund. Math. (12), 1928.
- Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
- R. Kaufman, Fourier transforms and descriptive set theory, Mathematika 31 (1984), no. 2, 336–339 (1985). MR 804207, DOI https://doi.org/10.1112/S0025579300012547 ---, private communication, January 1985. A. S. Kechris, A. Louveau, J. Saint-Raymond and J. Stern, Inaccessible cardinals and characterizations of Polish spaces (in preparation).
- Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Alain Louveau, Ensembles analytiques et boréliens dans les espaces produits, Astérisque, vol. 78, Société Mathématique de France, Paris, 1980 (French). With an English summary. MR 606933
- Alain Louveau, Recursivity and capacity theory, Recursion theory (Ithaca, N.Y., 1982) Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 285–301. MR 791064, DOI https://doi.org/10.1090/pspum/042/791064
- Donald A. Martin, Infinite games, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 269–273. MR 562614
- Jean Saint-Raymond, Caractérisation d’espaces polonais. D’après des travaux récents de J. P. R. Christensen et D. Preiss, Séminaire Choquet, 11e–12e années (1971–1973), Initiation à l’analyse, Exp. No. 5, Secrétariat Mathématique, Paris, 1973, pp. 10 (French). MR 0473133
- Jean Saint-Raymond, La structure borélienne d’Effros est-elle standard?, Fund. Math. 100 (1978), no. 3, 201–210 (French). MR 509546, DOI https://doi.org/10.4064/fm-100-3-201-210 R. M. Solovay, private communication, December 1983.
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© Copyright 1987
American Mathematical Society