In economics, a utility representation theorem is a theorem asserting that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). If the agent prefers A to B, we write . The set of all such preference-pairs forms the person's preference relation. Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function - a function u that assigns a real number to each option, such that if and only if . Not every preference-relation has a utility-function representation. For example, if the relation is not transitive (the agent prefers A to B, B to C, and C to A), then it has no utility representation, since any such utility function would have to satisfy , which is impossible. A utility representation theorem gives conditions on a preference relation, that are sufficient for the existence of a utility representation. Often, one would like the representing function u to satisfy additional conditions, such as continuity. This requires additional conditions on the preference relation. Definitions The set of options is a topological space denoted by X. In some cases we assume that X is also a metric space; in particular, X can be a subset an Euclidean space Rm, such that each coordinate in {1,...,m} represents a commodity, and each m-vector in X represents a possible consumption bundle. Preference relations A preference relation is a subset of . It is denoted by either or : The notation is used when the relation is strict, that is, means that option A is strictly better than option B. In this case, the relation should be irreflexive, that is, does not hold. It should also be asymmetric, that is, implies that not . The notation is used when the relation is weak, that is, means that option A is at least as good as option B (A may be equivalent to B, or better than B). In this case, the relation should be reflexive, that is, always holds. Given a weak preference relation