The Algebraic versus the Topological Approach to Additive Representations

It is proved that, under a nontriviality assumption, an additive function on a Cartesian product of connected topological spaces is continuous, whenever the preference relation, represented by this function, is continuous. The result is used to generalize a theorem of Debreu (1960) on additive representations, and to argue that the algebraic approach of Krantz, Luce, Suppes, & Tversky (1971, Foundations of Measurement) to additive conjoint measurement is preferable to the more customary topological approach. Applications are given to the representation of strength of preference relations, and to the characterization of subjective expected utility maximization.