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