In 1955, Bartle, Dunford, and Schwartz developed a theory of integration for scalar functions with respect to a \(\sigma \) -additive Banach-space-valued vector measure \(\mu _\infty \) defined on a \(\sigma \) -algebra of sets and used it to give an integral representation for weakly compact operators \(u : C(S) \to \mathcal {X},\) where S is a compact Hausdorff space and \(\mathcal {X}\) is a Banach space. About 15 years later, Lewis studied a Pettis-type weak integral of scalar functions with respect to a \(\sigma \) -additive vector measure \(\mu _\infty \) having range in a locally convex Hausdorff \(\mathcal {X}\) .