Representation of increasing convex functionals with countably additive measures

TL;DR

This paper provides new representation theorems for increasing convex functionals using countably additive measures, applicable to continuous and Borel measurable functions, under verifiable semicontinuity conditions.

Contribution

It introduces two types of representation results for increasing convex functionals, expanding the theoretical framework with practical assumptions.

Findings

Max-representation for functionals on continuous functions

Sup-representation for functionals on Borel measurable functions

Applicable under easy-to-verify semicontinuity conditions

Abstract

We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a sup-representation of functionals defined on spaces of real-valued Borel measurable functions. Our assumptions consist of sequential semicontinuity conditions which are easy to verify in different applications.