  We show that any decoherence functional $D$ can be represented by a spanning
vector-valued measure on a complex Hilbert space. Moreover, this representation
is unique up to an isomorphism when the system is finite. We consider the
natural map $U$ from the history Hilbert space $K$ to the standard Hilbert
space $H$ of the usual quantum formulation. We show that $U$ is an isomorphism
from $K$ onto a closed subspace of $H$ and that $U$ is an isomorphism from $K$
onto $H$ if and only if the representation is spanning. We then apply this work
to show that a quantum measure has a Hilbert space representation if and only
if it is strongly positive. We also discuss classical decoherence functionals,
operator-valued measures and quantum operator measures.
