The theory of expected utility derives conditions in which an ordering > of a set P of probability distributions, all defined on some measurable space (X, 1), can be represented by the numerical order of the expected values J udP of some " utility " function u on X. The set P may comprise only the simple distributions usually considered in texts on utility theory, but may include all discrete distributions, or continuous distributions of various kinds, or perhaps all the probability measures which can be defined on S. The class U of permissible utility functions may also be defined in various ways: it may include all those for which the expected values are defined and finite, or may be restricted to functions having some prescribed properties, such as boundedness, continuity or differentiability. Professor K. J. Arrow [2] has recently proposed a condition of a type new to this theory, namely that the ordering be continuous with respect to a certain metric in the set P of distributions. Such an assumption reflects the intuitively appealing notion that distributions which are " close together " in their assignment of probabilities to events in X should also be neighbours in the order of preference. But there are various plausible ways in which " close together " can be defined mathematically, and the choice among them has radical implications-which can be far from intuitive-for the existence and properties of utility functions. The particular metric considered by Arrow turns out to be unsatisfactory in certain respects; but his proposal suggests an interesting general method, which it is the object of the present paper to explore systematically. The following general statement of the problem of the existence of utility functions is adequate for the present discussion; note that the structure defined in this statement, together with the stated properties of the symbols X, Y, P, 1, U, and Sy will be assumed without special mention throughout the paper: