The structure of 𝜎-ideals of compact sets

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

doi:10.1090/S0002-9947-1987-0879573-9

The structure of $\sigma$-ideals of compact sets
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by A. S. Kechris, A. Louveau and W. H. Woodin

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

DOI: https://doi.org/10.1090/S0002-9947-1987-0879573-9

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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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