  Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed
with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is
a measure (not necessarily a point measure) which stabilizes; this means that
$\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For
bounded test functions $f$ on $R^d$, we give weak and strong laws of large
numbers for $\nu_n(f)$. The general results are applied to demonstrate that an
unknown set $A$ in $d$-space can be consistently estimated, given data on which
of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells,
answering a question raised by Khmaladze and Toronjadze. Further applications
are given concerning the Gamma statistic for estimating the variance in
nonparametric regression.
