Polar σ-Ideals of Compact Sets

Let E be a metric compact space. We consider the space K(E) of all compact subsets of E endowed with the topology of the Hausdorff metric and the space M(E) of all positive measures on E endowed with its natural w*-topology. We study σ-ideals of K(E) of the form I = Ip= {K ∈ K(E): μ(K) = 0, ∀ μ ∈ P} where P is a given family of positive measures on E. If M is the maximal family such that I = 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'.