Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

On a given cone (resp. vector space) we consider an initial topology and order induced by a family of linear operators into a second cone which carries a locally convex topology. We prove that monotone linear functionals on which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on . 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.