Vector-valued capacities
type: Item
instance of: scholarly article
title: Vector-valued capacities
zbMATH Open document ID: 0913.28009
DOI: 10.1007/BF02844370
author: R. M. Shortt
author: Emma D'Aniello
publication date: 1998-11-18T00:00:00Z
review text: Let \(X\) be a Polish space, \(\mathbb{P}(X)\) the power set of \(X\) and \(B\) a Banach lattice. The authors introduce the notion of vector-valued capacity; that is a monotone function \(I: \mathbb{P}(X)\to B\) \(\sigma\)-order continuous from below and on closed sets \(\sigma\)-order continuous from above. The authors transfer some results known in the real-valued case to the vector-valued one. It is proved that for a capacity \(I\) any analytic subset of \(X\) is \(I\)-capacitable (i.e., approximable from below by compact sets), that the outer measure of a Borel measure on \(X\) with values in a reflexive Banach lattice is a capacity, and, in connection with the ``marginal problem'', it is given an example of a capacity. This example is based on Theorem 2 of \textit{A. Hirshberg} and \textit{R. M. Shortt} [Proc. Am. Soc. 126, No. 6, 1669-1671 (1998; Zbl 0904.28011)].
reviewed by: Hans Weber
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 1203699
zbMATH Keywords: vector-valued capacities
zbMATH Keywords: marginal problem
zbMATH Keywords: Banach lattice
MaRDI profile type: Publication
cites work: Q3876158
cites work: Theory of capacities
cites work: A version of Strassen’s Theorem for vector-valued measures
cites work: Duality theorems for marginal problems
cites work: Q4080169
cites work: Q4764702
cites work: Q4049667
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cites work: Q5580575
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full work available at URL: https://doi.org/10.1007/bf02844370
OpenAlex ID: W2047939375
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 1203699