For a compact Hausdorff space $K$, we give descriptions of the dual of $C(K)^\delta$, the Dedekind completion of the Banach lattice $C(K)$ of continuous, real-valued functions on $K$.  We characterize those functionals which are $\sigma$-order continuous and order continuous, respectively, in terms of Oxtoby's category measures.  As applications, we give a purely topological characterization of hyper-Stonean spaces, and characterize those spaces $K$ for which $C(K)$ admits a strictly positive order continuous functional.