Mean Values for Random Sets

For a stationary random closed set Z in R^d, the volume density or specific volume was defined in Section 2.4 by

$$\bar V_d (Z) = {{\rm E} \; \lambda (Z \cap B) \over \lambda (B)}$$

((9.1))

, where $B \subset {\rm{R}}^d$ can be an arbitrary Borel set with 0 < λ(B) < ∞. This important parameter describes the mean volume of the random set per unit volume of the space. It is obtained by a double averaging, stochastic and spatial. The straightforward definition (9.1) has the advantage that it immediately exhibits λ(Z ∩ B)/λ(B) as an unbiased estimator for the specific volume. The situation becomes less simple if one wants to take other quantitative aspects of point sets into account. For example, in several applications one is interested in the mean surface area (the mean perimeter in the plane) per unit volume. Clearly, one cannot just proceed as in the case of (9.1), since the surface area of Z ∩ B is in general not defined. Evidently, we must restrict the realizations of the random set Z as well as the ‘observation window’ B. For that reason, we shall assume in the following that the realizations of the closed random set Z belong to the extended convex ring S, the sets of which have the property that the intersection with any convex body is a finite union of convex bodies. Moreover, the observation window will be a compact convex set W with positive volume. In that case, Z ∩ W has a well-defined surface area. However, part of it generally comes from Z ∩ bd W and not from the boundary of Z. To overcome boundary effects caused by the window W, the definition of densities for functionals other than the volume will require additional devices, for example, limit procedures.