This section describes a general theorem for additive measurement in Fishburn (1992b) and applies it to the closed-interval representation
where and . A sample of other approaches to interval and more general threshold representations is provided in Doignon et al. (1986), Chateauneuf (1987), Nakamura (1988), Suppes et al. (1989, Chapter 16), Beja and Gilboa (1992), Abbas and Vincke (1993), Bogart and Trenk (1994), Abbas (1995) and Mitas (1995).
Our theorem is a linear separation theorem for arbitrary systems of linear inequalities that have finite numbers of terms. We begin with a nonempty set Y, and let V denote the vector space of all for which is finite. We define and v + v' for real and by
The representation for the theorem consists of distinguished subsets A and B of V and a mapping for which
where . We say that (A,B) is solvable if there exists a that satisfies linear system (10). We will state a condition on (A,B) that is necessary and sufficient for solvability. It is assumed, with no loss of generality, that the zero function 0 of V is in A and that B is not empty.
A few other definitions are needed. For , . A subset K of V is a convex cone if it is nonempty, closed under convex combinations, and contains whenever and . A convex cone K is without origin if . The convex cone generated by nonempty is denoted by , so
Finally, we say that nonempty is Archimedean if for all , for some .
Our separation theorem says that (A,B) is solvable if and only if is included in some Archimedean convex cone without origin in V. Given (10), necessity of the condition on is shown by extending linearly to all of V by and observing that is an Archimedean convex cone without origin that includes . The sufficiency proof is based on a standard separation theorem discussed, for example, in Kelley and Namioka (1963) and Klee (1969).
To apply the theorem to the opening representation of this section, let X' be a disjoint copy of X with corresponding to , and let . The opening representation can then be rewritten as
Sets A and B for application of the separation theorem are
Suppose is included in an Archimedean convex cone without origin. Let satisfy (10). Then, when and , and , or and ; for , , or ; when , , or .