Riesz representation theorems for positive linear operators

We generalise the Riesz representation theorems for positive linear functionals on $$\text {C}_{\text {c}}(X)$$ and {0}}(X)$$ , where X is a locally compact Hausdorff space, to operators from these spaces into partially ordered vector space E. The representing measures are defined Borel $$\sigma $$ -algebra of take their values in extended cone corresponding integrals order integrals. give explicit formulas at open subsets X. Results included E need not be lattice, nor normed space. Representing exist, example, Banach lattices with continuous norms, regular KB-spaces, self-adjoint complex Hilbert spaces, JBW-algebras.