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.
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Roth, W. Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies. Positivity 6, 115–127 (2002). https://doi.org/10.1023/A:1015236803527
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DOI: https://doi.org/10.1023/A:1015236803527