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Transactions of the American Mathematical Society

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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.


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  • [B] N. Bary, A treatise on trigonometric series, vols. 1 and 2, Macmillan, 1964. MR 0171116 (30:1347)
  • [BM] J. P. Burgess and R. D. Mauldin, Conditional distributions and orthogonal measures, Ann. Proba. 9 (1981), 902-906. MR 628885 (82j:28002)
  • [CM] D. Cenzer and R. D. Mauldin, Faithful extensions of analytic sets to Borel sets, Houston J. Math. 6 (1980), 19-29. MR 575911 (82a:54071)
  • [Ch] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1955), 131-295. MR 0080760 (18:295g)
  • [Chr] J. P. R. Christensen, Necessary and sufficient conditions for the measurability of certain sets of closed subsets, Math. Ann. 200 (1973), 189-193. MR 0334169 (48:12488)
  • [D1] C. Dellacherie, Ensembles analytiques, capacités, mesures de Hausdorff, Lecture Notes in Math., vol. 295, Springer-Verlag, Berlin and New York, 1972. MR 0492152 (58:11301)
  • [D2] -, Capacités et processus stochastiques, Ergeb. Math. 67, Springer-Verlag, 1972. MR 0448504 (56:6810)
  • [DFM] C. Dellacherie, D. Feyel and G. Mokobodzki, Intégrales de capacités fortement sous-additives, Sém. Prob. Strasbourg XVI, Lecture Notes in Math., vol. 920, Springer-Verlag, 1982, pp. 8-28. MR 658670 (84f:31016)
  • [D3] C. Dellacherie, Appendice á l'exposé précédent, ibid., pp. 29-40.
  • [DM] C. Dellacherie and P. A. Meyer, Probabilités et potentiel, vol. 3, Hermann, Paris, 1984. MR 0488194 (58:7757)
  • [Hi] G. Hillard, Une généralisation du théorème de Saint-Raymond sur les boréliens à coupes $ {K_\sigma}$, C.R. Acad. Sci. Paris 288 (1970), 749-751. MR 535803 (80d:28003)
  • [Hu] W. Hurewicz, Relativ Perfecte Teile von Punktmengen und Mengen (A), Fund. Math. (12), 1928.
  • [KS] J. P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Hermann, Paris, 1963. MR 0160065 (28:3279)
  • [K1] R. Kaufman, Fourier transforms and descrpitive set theory, Mathematika 31 (1984), 336-339. MR 804207 (87f:42024)
  • [K2] -, private communication, January 1985.
  • [KLSS] A. S. Kechris, A. Louveau, J. Saint-Raymond and J. Stern, Inaccessible cardinals and characterizations of Polish spaces (in preparation).
  • [KS] J. P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Hermann, Paris, 1963. MR 0160065 (28:3279)
  • [Ku] K. Kuratowski, Topology, vols. 1 and 2, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [L1] A. Louveau, Ensembles analytiques et boréliens dans les espaces produits, Astérisque 78 (1980). MR 606933 (82j:03062)
  • [L2] -, Recursivity and capacity theory, Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, R.I., 1985, pp. 285-301. MR 791064 (87h:28004)
  • [MK] D. A. Martin and A. S. Kechris, Infinite games and effective descriptive set theory, Analytic Sets (C. A. Rogers et al., Eds.), Academic Press, 1980, pp. 404-469. MR 562614 (81i:04003)
  • [StR1] J. Saint Raymond, Caractérisations d'espaces Polonais, d'après des travaux recents de J.P.R. Christensen et D. Preiss, Sém. Choquet 11$ ^{e}$-12$ ^{e}$ annés, 1971-1973, no.5. MR 0473133 (57:12811)
  • [StR2] -, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1978), 201-210. MR 509546 (80g:54044)
  • [S] R. M. Solovay, private communication, December 1983.

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DOI: https://doi.org/10.1090/S0002-9947-1987-0879573-9
Article copyright: © Copyright 1987 American Mathematical Society

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