Point processes and completely monotone set functions

Abstract

Inherent in the proof of Renyi's Theorem [9] (as observed by Kallenberg [4] and M6nch [8]) is the fact that the distribution of an orderly point process (a process without multiple points) is determined by the zero probability function 4)(B)--P{N(B)=O} (N(B)=number of points in B). It is natural to ask what set functions ~b can arise as zero probability functions of point processes. In this paper we characterize these set functions in terms of a property that can naturally be called complete monotonicity. In his classical paper on capacities [1], Choquet studies a class of set functions which he calls alternating capacities of order infinity. Based on the integral representation of these functions in terms of extremal functions, Choquet observes that a class of these functions can be interpreted as giving the probability that sets intersect a random compact set. The extremal functions are of the form