Outer Minkowski content for some classes of closed sets

We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in the d-dimensional Euclidean space, in connection with regularity properties of their boundaries. Moreover, we provide a class of sets stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class. We find analogous conditions, stable under finite unions as well, for the existence of the mean outer Minkowski content of random closed sets. Finally, an application to birth-and-growth stochastic processes is briefly discussed.