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.
Highlights
When treating utility theory, traditional economic textbooks discuss two disparate cases in considerable detail: the potential non-existence of utility functions for complete and transitive preference relations on non-trivial connected Euclidean domains—usually illustrated by lexicographic preferences (Debreu, [1])—and the existence of continuous utility functions for complete, transitive and continuous preferences on connected Euclidean domains; see, e.g. Mas-Colell, Whinston, and Green [2]
The outer measure is defined likewise as the infimum over coverings whose sizes have been defined. We follow this approach to define the utility of an alternative as the outer measure of its set of worse alternatives
To the best of our knowledge, the logical connection between outer measure and utility has never been made before. We hope that this link between utility theory and measure theory is more explicit, intuitive and mathematically elementary than the above-mentioned approaches
Summary
Traditional economic textbooks discuss two disparate cases in considerable detail: the potential non-existence of utility functions for complete and transitive preference relations on non-trivial connected Euclidean domains—usually illustrated by lexicographic preferences (Debreu, [1])—and the existence of continuous utility functions for complete, transitive and continuous preferences on connected Euclidean domains; see, e.g. Mas-Colell, Whinston, and Green [2]. The outer measure is the best such approximation “from above” This is illustrated in Figure 1: having defined the size of rectangles in the plane, we can assign a size to more general sets S in the plane by covering it with rectangles. We follow this approach to define the utility of an alternative as the outer measure of its set of worse alternatives. To the best of our knowledge, the logical connection between outer measure and utility has never been made before We hope that this link between utility theory and measure theory is more explicit, intuitive and mathematically elementary than the above-mentioned approaches. The utility function in terms of outer measure is simple and intuitive, it delivers the most general results possible.
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