The structure of \(\sigma\)-ideals of compact sets. (English) Zbl 0633.03043

Kechris, A. S.; Louveau, A.; Woodin, W. H.

Trans. Am. Math. Soc. 301, 263-288 (1987).

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.

MSC:

03E15	Descriptive set theory
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
28A12	Contents, measures, outer measures, capacities
42A63	Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
54D30	Compactness
54E35	Metric spaces, metrizability

Keywords:

\(\sigma \)-ideals of compact sets; compact metrizable spaces; complexity; dichotomy theorem; generators; Borel bases; definability; countable chain condition