Mean density of inhomogeneous Boolean models with lower dimensional typical grain

The mean density of a random closed set $\Theta$ in $\R^d$ with Hausdorff dimension $n$ is the Radon-Nikodym derivative of the expected measure $\E[\h^n(\Theta\cap\cdot)]$ induced by $\Theta$ with respect to the usual $d$-dimensional Lebesgue measure. We consider here inhomogeneous Boolean models with lower dimensional typical grain. Under general regularity assumptions on the typical grain, related to the existence of its Minkowski content, and on the intensity measure of the underlying Poisson point process, we prove an explicit formula for the mean density. The proof of such formula provides as by-product estimators for the mean density in terms of the empirical capacity functional, which turns to be closely related to the well known random variable density estimation by histograms in the extreme case $n=0$. Particular cases and examples are also discussed.