On the approximation of mean densities of random closed sets

Many real phenomena may be modeled as random closed sets in R, of different Hausdorff dimensions. 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 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.