  Mean density of lower dimensional random closed sets, as well as the mean
boundary density of full dimensional random sets, and their estimation are of
great interest in many real applications. Only partial results are available so
far in current literature, under the assumption that the random set is either
stationary, or it is a Boolean model, or it has convex grains. We consider here
non-stationary random closed sets (not necessarily Boolean models), whose
grains have to satisfy some general regularity conditions, extending previous
results. We address the open problem posed in (Bernoulli 15 (2009) 1222-1242)
about the approximation of the mean density of lower dimensional random sets by
a pointwise limit, and to the open problem posed by Matheron in (Random Sets
and Integral Geometry (1975) Wiley) about the existence (and its value) of the
so-called specific area of full dimensional random closed sets. The
relationship with the spherical contact distribution function, as well as some
examples and applications are also discussed.
