Convex Functions on σ-Algebras of Nonatomic Measure Spaces

摘要

The purpose of this paper is to present a convex-like structure of set functions on σ-algebras of nonatomic finite measure spaces. A key notion relating to the convexity of families of measurable sets and set functions defined on such families is that of convex combinations of measurable sets. We prove Jensen inequality and show that the set of minimizers of convex set functions is convex. We then metrize σ-algebras and study the continuity of set functions on σ-algebras as continuous functions on metric spaces. Specifically, we prove a minimax theorem for set functions and investigate how the convexity and the absolute continuity of set functions, and the continuity and the countable additivity of finitely additive set functions, are mutually related. The main results are applied to study the properties of cores of cooperative games with transferable utility and the existence of a fair division.