Local spherical contact distribution function and local mean densities for inhomogeneous random sets

In this paper we provide definitions for the local mean volume and mean surface densities of an inhomogeneous random closed set. A theorem which relates the local spherical contact distribution function with the local surface and volume density is proven. Sufficient conditions on the regularity of the random set involved to satisfy the assumptions of the theorem are provided, based on Coarea Formula. These conditions are satisfied by a wide class of inhomogeneous random sets, relevant for applications, like some kinds of Boolean Models, for which explicit expressions for the local volume and surface densities are also provided.