POLAR ff-IDEALS OF COMPACT SETS

Let £ be a metric compact space. We consider the space 3T{E) of all compact subsets of E endowed with the topology of the Hausdorff metric and the space Jt{E) of all positive measures on E endowed with its natural w*-topology. We study c-ideals of 5F(E) of the form I = Ip = {K e 5P(E): n(K) = 0,V^SP} where P is a given family of positive measures on E. If M is the maximal family such that / = Im , then M is a band. We prove that several descriptive properties of I: being Borel, and having a Borel basis, having a Borel polarity-basis, can be expressed by properties of the band M or of the orthogonal band M'. Although this work will not use much Harmonic Analysis, its motivation goes back to the study of the interesting a -ideals U and Uo on the torus T. Recall that a compact subset K of T is a £/-set if and only if K does not support any distribution S such that lim,,-^ \S(n)\ = 0 (where S is the Fourier transform of S). The notion of compact {70-set is obtained just by replacing in the last definition "distribution" by "measure" or equivalently by "positive measure". It has been well known since the fundamental work of Pyateskii-Sapiro that U is a proper subset of Uo. However, recent results, of A. Kechris and A. Louveau (see [7]), and J. Saint Raymond and the author (see [1]) have shown some descriptive structural properties distinguishing these two <r-ideals. For example, Uq has a Borel basis whereas U does not. Another property—call it (*)—satisfied by U0 and not by U is the following: Given any Borel subset A of T, either A is contained in the union of a countable family of compact Co-sets or A contains a compact set not in Uq . (For undefined notions and more details see [1, 4, 7]). From now on, E denotes a metric compact space and J£{E) the space of all compact subsets of E endowed with the Hausdorff topology. The properties of Co mentioned above are shared by the rr-ideal 3£m{E) of all countable compact subsets of E. In fact, the family of all singletons forms a closed basis for 3?m{E) and the property (*) is just a reformulation of the classical "perfect set theorem" for Borel sets. The starting point of our work is the simple observation that, unlike U, both a-ideals f/0 and 3?W{E) are of the form I = IP = {K € X{E): n(K) = 0, V/ie?} where P is some fixed family of measures. This follows from the definition in the case of Uo, and Received by the editors May, 19, 1993 and, in revised form, April 1, 1994. 1991 Mathematics Subject Classification. Primary 28A99, 46A55, 04A15; Secondary 42A63. ©1995 American Mathematical Society