Completeness of Decoherence Functionals

The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional $d$. In its recent formulation by Isham it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. When $\UP$ is given by the lattice of projectors $\PV$ in some Hilbert space $\V$, consistent sets correspond to certain partitions of the unit operator in $\V$ into mutually orthogonal projectors $\{\a_1,\a_2,\ldots\}$, such that the function $d(\a,\a)$ is a probability distribution on the boolean algebra generated by $\{\a_1,\a_2,\ldots\}$. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where $\V$ is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution $p(\a)$ on the corresponding boolean algebra, decoherence functionals $d$ with respect to which this set is consistent and which are such that for the probability functions $d(\a,\a)=p(\a)$ holds.