An Elementary Proof That Well-Behaved Utility Functions Exist

Abstract

Starting from an intuitive and constructive approach for countable domains, and combining this with elementary measure theory, we obtain an upper semi-continuous utility function based on outer measure. Whenever preferences over an arbitrary domain can at all be represented by a utility function, our function does the job. Moreover, whenever the preference domain is endowed with a topology that makes the preferences upper semi-continuous, so is our utility function. Although links between utility theory and measure theory have been pointed out before, to the best of our knowledge, this is the first time that the present intuitive and straight-forward route has been taken.