Vector-valued capacities (Q1265438)

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)].