Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
MathSciNet review: 879573
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03E15, 28A05, 42A63

Retrieve articles in all journals with MSC: 03E15, 28A05, 42A63


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0879573-9
PII: S 0002-9947(1987)0879573-9
Article copyright: © Copyright 1987 American Mathematical Society