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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The structure of $\sigma$-ideals of compact sets
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by A. S. Kechris, A. Louveau and W. H. Woodin PDF
Trans. Amer. Math. Soc. 301 (1987), 263-288 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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