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We prove that, in a measure-theoretic sense, given a fixed radius $r$, the maximum density acheived by packings of the hyperbolic plane by radius $r$ circles is the supremum of densities of ``periodic packings'' (those packings with cofinite symmetry). We also show that the maximal density function is continuous (as a function of radius). Our investigations lead us to consider more general questions regarding the density of periodic measures in a space of measures naturally defined for any group.