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 $\text{UP}$ of propositions about histories with an orthoalgebra or lattice. When $\text{UP}$ is given by the lattice of projectors $\text{PV}$ in some Hilbert space $\text{V}$, consistent sets correspond to certain partitions of the unit operator in $\text{V}$ into mutually orthogonal projectors $\{{\text{a}}_1,{\text{a}}_2,\dots \}$, such that the function $d(\text{a},\text{a})$ is a probability distribution on the boolean algebra generated by $\{{\text{a}}_1,{\text{a}}_2,\dots \}$. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where $\text{V}$ 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 $p(\text{a})$ 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 $d(\text{a},\text{a})=p(\text{a})$ holds.