Polar $\sigma$-ideals of compact sets

Abstract:

Let $E$ be a metric compact space. We consider the space $\mathcal {K}(E)$ of all compact subsets of $E$ endowed with the topology of the Hausdorff metric and the space $\mathcal {M}(E)$ of all positive measures on $E$ endowed with its natural ${w^{\ast }}$-topology. We study $\sigma$-ideals of $\mathcal {K}(E)$ of the form $I = {I_P} = \{ K \in \mathcal {K}(E):\mu (K) = 0,\;\forall \mu \in P\}$ where $P$ is a given family of positive measures on $E$. If $M$ is the maximal family such that $I = {I_M}$, 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 $M’$.