A simple mathematical result characterizing a subset of a product set is proved and used to obtain additive representations of preferences. The additivity consequences of independence assumptions are obtained for preferences which are not total or transitive. This means that most of the economic theory based on additive preferences - expected utility, discounted utility - has been generalized to preferences which are not total or transitive. Other economic applications of the theorem are given.
(This abstract was borrowed from another version of this item.)