  Given $n$ independent random marked $d$-vectors $X_i$ with a common density,
define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not
necessarily a point measure) determined by the (suitably rescaled) set of
points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a
suitable power-law decay of the tail of the radius of stabilization. For
bounded test functions $f$ on $R^d$, we give a law of large numbers and central
limit theorem for $\nu_n(f)$. The latter implies weak convergence of
$\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on
bounded test functions. The general result is illustrated with applications
including the volume and surface measure of germ-grain models with unbounded
grain sizes.
