Concepts of similarity for utility functions

A generalization of the compact-open topology is defined for a space of utility functions with different choice sets. The space is a complete separable metric space. A continuous representation theorem of Levin (1983) gives a homeomorphism between this space and the space of preference relations, topologized by closed convergence. A map into the space is measurable with regard to the Borel algebra iff the choice set correspondence is measurable and the utility function is measurable on its graph. A coarser topology (that of subgraph convergence) is also studied. This topology is coarser than the compact-open topology when the utility functions are defined on the same choice set. However the demand correspondence is still upper-hemicontinuous. A homeomorphism is given between this space and the space of preference relations, with the latter given a certain topology coarser than closed convergence.