Toward a fundamental theorem of quantal measure theory

In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice Z(n). In both cases, the dynamics can be coded into a vector-valued measure mu on Omega, the space of all histories. Initially, mu is just defined on special subsets of Omega called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally mu is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.