Integration theory of observables

Representations of abstract observables on a generalised logic are given in terms of bounded vector-valued Borel measures on the real line whose ranges are in the dual spaceX^* of the Banach space of statesX. Each bounded observable is furthermore represented by an elementu^* ofX^* such that for any proper statep X, u^*(p) is the expectation value ofu when the system is in the statep.