$G_{\delta }$ ideals of compact sets

Abstract

We investigate the structure of Gδ ideals of compact sets. We define a class of Gδ ideals of compact sets that, on the one hand, avoids certain phenomena present among general Gδ ideals of compact sets and, on the other hand, includes all naturally occurring Gδ ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all Gδ ideals. In particular, we establish a result representing ideals in this class via the meager ideal. This result is analogous to Choquet's theorem representing alternating capacities of order ∞ via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing important to us examples of Gδ ideals outside of our class.