Some remarks on the continuity of random closed sets

A concept of absolutely continuous random closed set has been recently introduced by the authors, in terms of the absolute continuity of a suitable mean content measure associated with its boundary, with respect to the usual Lebesgue measure. It is shown that our definition of continuity, respectively absolute continuity, of random closed sets can be seen as an extension of the standard definition of continuous, respectively absolutely continuous, random variables (or random vectors); in particular absolute continuity implies continuity, in our stochastic geometric context too. It is then compared with definitions of continuity of random closed sets that have already appeared in the relevant literature. Examples show how the definition proposed by the present authors is of interest in many significant applications.