\(G_{\delta }\) ideals of compact sets (Q550555)

Summary: We investigate the structure of \(G_{\delta }\) ideals of compact sets. We define a class of \(G_{\delta }\) ideals of compact sets that, on the one hand, avoids certain phenomena present among general \(G_{\delta }\) ideals of compact sets and, on the other hand, includes all naturally occurring \(G_{\delta }\) ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all \(G_{\delta }\) 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 \(\infty \) via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing examples of \(G_{\delta }\) ideals outside of our class that are important to us.