On the upper semicontinuity of Choquet capacities
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.
scientific article; zbMATH DE number 5758589
scientific article