  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 defined only on special subsets of Omega called cylinder-events, and one
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 which in a suitable sense converges to the event whose
measure one is 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.
