Some results on the existence of utility functions on path connected spaces

Abstract

This paper concerns the existence of utility representations for preferences defined on a path connected space X. This includes any convex set. A classical result of Eilenberg (1941) proves the existence of utility representations when the consumption set is connected and separable. In an infinite dimensional space the above result may not be useful, because we lack, in general, the separability of the space. The non-separable spaces L ∞ and ca( K) are typical examples in mathematical economics. In this paper we show that a continuous preference relation ≽, on X has a continuous utility representation if and only if it is countably bounded, i.e., there is some countable subset F of X such that for all x in X there exist y and z in F with y≽ x≽ z. An easy corollary states that any continuous preference which has a best and a worst point has a continuous representation. We also obtain a convex continuous preference on a Banach lattice that has not a utility representation, because it is not countably bounded.