Locally Finite Random Sets: Foundations for Point Process Theory

Abstract

The foundations of point process theory are surveyed. An abstract theory motivated by applications in stochastic geometry is presented. It is shown that it is sufficient to know only which sets are measurable and which are bounded in the basic space, where we use countability hypotheses rather than topological assumptions. (The sole exception is in the construction of probabilities where pseudo-topological hypotheses are needed.) It is shown that there are close connections with the random set theories of Kendall and Matheron.