On the approximation of geometric densities of random closed sets

Many real phenomena may be modelled as random closed sets in Rd, of different Hausdorff dimensions. The authors have recently revisited the concept of mean geometric densities of random closed sets θn with Hausdorff dimension n ≤ d with respect to the standard Lebesgue measure on Rd, in terms of expected values of a suitable class of linear functionals (Delta functions à la Dirac). In many real applications such as fiber processes, n-facets of random tessellations of dimension n ≤ d in spaces of dimension d ≥ 1, several problems are related to the estimation of such mean densities; in order to face such problems in the general setting of spatially inhomogeneous processes, we suggest and analyze here an approximation of mean densities for sufficiently regular random closed sets. We will show how some known results in literature follow as particular cases. A series of examples throughout the paper are provided to exemplify various relevant situations.