We prove that given a fixed radius $r$, the set of isometry-invariant probability
measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is
dense in the space of all isometry-invariant probability measures on the space of radius
$r$-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a
corollary, we prove the maximum density achieved by isometry-invariant probability measures
on a space of radius $r$-packings of the hyperbolic plane is the supremum of densities of
periodic packings. We also show that the maximum density function varies continuously with
radius.