Abstract 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.
Event structures in nonstandard quantum mechanics are represented by closed subspaces of a nonstandard Hilbert space. There are four natural types of closed subspaces of a nonstandard Hilbert space and these different types coalesce in the standard theory. These types are the closed internal subspaces, the splitting subspaces, the ⊥-closed subspaces, and the norm-closed subspaces. The algebraic properties of these four types are studied and compared. Moreover, the resulting structures are compared to others that have recently appeared in the literature.