  The standard (``fine-grained'') interpretation of quantum density functional
theory, in which densities are specified with infinitely-fine spatial
resolution, is mathematically unruly. Here, a coarse-grained version of DFT,
featuring limited spatial resolution, and its relation to the fine-grained
theory in the $L^1\cap L^3$ formulation of Lieb, is studied, with the object of
showing it to be not only mathematically well-behaved, but consonant with the
spirit of DFT, practically (computationally) adequate and sufficiently close to
the standard interpretation as to accurately reflect its non-pathological
properties. The coarse-grained interpretation is shown to be a good model of
formal DFT in the sense that: all densities are (ensemble)-V-representable; the
intrinsic energy functional $F$ is a continuous function of the density and the
representing external potential is the (directional) functional derivative of
the intrinsic energy. Also, the representing potential $v[\rho]$ is
quasi-continuous, in that $v[\rho]\rho$ is continuous as a function of $\rho$.
The limit of coarse-graining scale going to zero is studied to see if
convergence to the non-pathological aspects of the fine-grained theory is
adequate to justify regarding coarse-graining as a good approximation. Suitable
limiting behaviors or intrinsic energy, densities and representing potentials
are found. Intrinsic energy converges monotonically, coarse-grained densities
converge uniformly strongly to their low-intrinsic-energy fine-grainings, and
$L^{3/2}+L^\infty$ representability of a density is equivalent to the existence
of a convergent sequence of coarse-grained potential/ground-state density
pairs.
