Scott functions, their representations on domains, and applications to random sets

Abstract

Choquet theorems (1954) on integral representation for capacities are fundamental to probability theory. They inspired a growing body of research into different approaches and generalizations of Choquet's results by many other researchers. Notably Mathéron's work (1975) on distributions over the space of closed subsets has led to further advancements in the theory of random sets. This paper was inspired by the work of Norberg (1989) who generalized Choquet's results to distributions over domains. While Choquet's original theorems were obtained for locally compact Hausdorff (LCH) spaces, both Mathéron's and Norberg's depend on the assumption of separability in their application of the Carathéodory's method. Our Radon measure approach differs from the work of Mathéron and Norberg, in that it does not require separability. This investigation naturally leads to the introduction of finite and locally finite valuations, which allows us to characterize finite and locally finite random sets in terms of capacities on the class of compact subsets. Finally, the treatment of Lévy exponent by Mathéron and Norberg is revisited, and the notion of exponential valuation is proposed for the representation of general Poisson processes.