ABSTRACT
This chapter is concerned with the foundations of random sets taking values as closed subsets of Rd (or more generally of a Hausdorff, locally compact and second countable topological space).
In Chapters 3 and 4 we considered random sets on a finite set U , i.e., random elements taking subsets of U as values. These correspond to games of chance in classical probability theory. While the theory of discrete random variables (or vectors), i.e., random elements with values in an at most countable subset D of Rd, follow in a straightforward manner from finite random variables, the situation is different for random sets. This is so because if U is discrete (infinitely countable, e.g., the set of integers Z), its power set 2U is infinitely uncountable, and its power set (2U ,⊆) is not locally finite. Note that we are talking about random sets on a discrete space U , i.e., random elements with values in 2U . If we consider discrete random sets, i.e., random sets taking values in a discrete subset D of 2U , then of course, their probability laws are determined by their densities on D. Specifically, let X be a discrete random set on U , with density f on D (i.e., f : D → [0, 1], ∑
A∈D f(A) = 1), then
∀A ⊆ 2U , P (X ∈ A) =
∑ A∈D∩A
f(A).
