Riesz representation theorems for vector lattices and Banach lattices of regular operators

For a non-empty locally compact Hausdorff space X and a Dedekind complete normal vector lattice E, we show that the vector lattice of norm to order bounded operators from Cc(X) or C₀(X) into E is isomorphic to the vector lattice of E-valued regular Borel measures on X. When E is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When X is compact, every regular operator from C(X) into E is norm to order bounded. For some spaces E, such as KB-spaces or the regular operators on a KB-space, every regular operator from C₀(X) into E is norm to order bounded. Additional results are obtained for the whole space of regular operators from Cc(X) into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice E, resp. in the extended positive cone of E, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When E is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of Cc(X) and C₀(X).