Distributions of random closed sets via containment functionals (Q2925030)

From the author's introduction: A key result in the theory of random sets, the Choquet theorem, states that the distribution of a random closed set \(X\) is characterized by its hitting functional \(T_X: K \mapsto P(X\cap K\neq \emptyset)\), where \(K\) is compact. In the usual presentation, the carrier space is locally compact, second countable and Hausdorff (LCSH). \textit{T. Norberg} [Math. Scand. 64, No. 1, 15--51 (1989; Zbl 0667.60001)] showed that Hausdorffness can be replaced by the weaker property of sobriety. Another approach is to try a similar characterization for the containment functional \(C_X:\) \(F\mapsto P(X\subset F)\), where \(F\) is closed. Although equivalent in LCSH space, the two approaches depart in more general spaces. NEWLINENEWLINENEWLINEThis paper presents such a characterization of distributions by identifying their containment functionals as the completely monotone, outer continuous capacities. The result is valid in any Hausdorff space which is both locally compact and \(\sigma\)-compact, thus dropping the assumption of second countability. NEWLINENEWLINENEWLINEThe structure of the paper is as follows. Section 2 collects the basic notation and definitions. The main result is presented in Section 3 and proved in Section 4 and 5. Section 6 shows examples of random closed sets in spaces without the second countability axiom. Section 7 closes the paper with some final remarks.