CONVEX FUNCTIONS ON σ-ALGEBRAS OF NONATOMIC MEASURE SPACES

The purpose of this paper is to present a convex-like structure of set functions on sigma-algebras of nonatomic finite measure spaces Using the nonatomicity of measures, we introduce a convex subset of a sigma-algebra, a mu-convex set, mid it convex set function, a mu-convex function, in a reasonably standard way analogous to convex analysis We prove Jensen inequality for mu-convex functions and show that the set of minimizers of mu-convex functions is mu-convex We then metrize sigma-algebras and study the continuity of set functions on sigma-algebras as continuous functions on metric spaces Specifically, we prove a minimax theorem for set functions and investigate how the mu-convexity and the absolute continuity of set functions, and the continuity and the countable additivity of finitely additive set functions, are mutually related