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Polar $ \sigma$-ideals of compact sets


Author: Gabriel Debs
Journal: Trans. Amer. Math. Soc. 347 (1995), 317-338
MSC: Primary 28A12; Secondary 04A15, 28A15, 46A55
DOI: https://doi.org/10.1090/S0002-9947-1995-1267222-4
MathSciNet review: 1267222
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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'$.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1267222-4
Article copyright: © Copyright 1995 American Mathematical Society

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