  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. This leads to a purely topological
characterization of hyper-Stonean spaces.
