  We point out that a quantum system with a strongly positive quantum measure
or decoherence functional gives rise to a vector valued measure whose domain is
the algebra of events or physical questions. This gives an immediate handle on
the question of the extension of the decoherence functional to the sigma
algebra generated by this algebra of events. It is on the latter that the
physical transition amplitudes directly give the decoherence functional. Since
the full sigma algebra contains physically interesting questions, like the
return question, extending the decoherence functional to these more general
questions is important. We show that the decoherence functional, and hence the
quantum measure, extends if and only if the associated vector measure does. We
give two examples of quantum systems whose decoherence functionals do not
extend: one is a unitary system with finitely many states, and the other is a
quantum sequential growth model for causal sets. These examples fail to extend
in the formal mathematical sense and we speculate on whether the conditions for
extension are unphysically strong.
