On the upper semicontinuity of Choquet capacities
type: Item
instance of: scholarly article
title: On the upper semicontinuity of Choquet capacities
zbMATH Open document ID: 1205.60031
author: Guo Wei
author: Yangeng Wang
author: Donald E. Beken
author: Hung T. Nguyen
published in: International Journal of Approximate Reasoning
publication date: 2010-07-20T00:00:00Z
review text: The distribution of a random closed set \(X\) in a locally compact second countable Hausdorff space \(E\) is uniquely determined by its capacity functional \(T(K)=\mathbf{P}(X\cap K\neq\emptyset)\) for all \(K\) from the family \(\mathcal{K}\) of compact sets. It is well known that the capacity functional is upper semicontinuous on \(\mathcal{K}\), equivalently that \(T(K_n)\downarrow K\) for each sequence of compact sets such that \(K_n\downarrow K\). The capacity functional can be extended to the family of closed sets, however, it is no longer upper semicontinuous on this family. The authors use the one-point compactification \(\bar{E}\) of \(E\) and work with the random closed set \(\bar{X}\) there which corresponds to (induced by) \(X\). They note that the capacity functional of \(\bar{X}\) is upper semicontinuous in the compactified space and list various consequences of this fact.
reviewed by: Ilya S. Molchanov
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 5758589
zbMATH Keywords: Choquet capacity
zbMATH Keywords: random closed set
zbMATH Keywords: continuity from above
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: compactification
MaRDI profile type: Publication
full work available at URL: j.ijar.2009.12.002
OpenAlex ID: W1980023388
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DOI: 10.1016/J.IJAR.2009.12.002
description: scientific article; zbMATH DE number 5758589