| The purpose of this paper is to present a convex-like structure of set functions on σ-algebras of nonatomic finite measure spaces. Using the nonatomicity of measures, we introduce a convex subset of a σ-algebra, a μ-convex set, and a convex set function, a μ-convex function, in a reasonably standard way analogous to convex analysis. We prove Jensen inequality for μ-convex functions and show that the set of minimizers of μ-convex functions isμ-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 μ-convexity and the absolute continuity of set functions, and the continuity and the countable additivity of finitely additive set functions, are mutually related. |
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