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

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 |