Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $ν_n=\sum_iξ_i$, where $ξ_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $ξ_i$ 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 $ν_n(f)$. 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.