Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

Abstract

On a given cone (resp. vector space) $\mathcal{Q}$ we consider an initial topology and order induced by a family of linear operators into a second cone $\mathcal{P}$ which carries a locally convex topology. We prove that monotone linear functionals on $\mathcal{Q}$ which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on $\mathcal{P}$. Based on the Riesz representation theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces.