Representation of preference relations on σ-algebras of nonatomic measure spaces: Convexity and continuity

The purpose of this paper is to show how preference relations on σ-algebras can be represented by means of nonadditive set functions that satisfy appropriate requirements of convexity and continuity on σ-algebras. To this end, we introduce convex combinations of measurable sets, and quasiconcave and concave functions on σ-algebras, which conform with the standard results in convex analysis. We formulate the convexity and the continuity axioms for preference relations on σ-algebras with a metric topology and show the existence of utility functions for convex continuous preference relations. We also show that monotone continuous preference relations are representable by fuzzy measures.