Counting measures,monotone random set functions

Summary This paper arose from work on random processes whose values are measures or more general set functions. Secs. 1–3, which have nothing specifically random, discuss two topologies for certain sigma-finite measures. One, applicable only to counting measures, is a quotient topology which is useful in the finite case but excessively weak in the infinite case. Making use of a well-known result of P. Hall on sets of representatives, we describe this topology and show that it can be enlarged to the stronger one generated by a modification of the Lévy-Prohorov (L-P) metric. Sec. 4 gives a property of the L-P metric for finite integer valued counting measures. The rest of the paper deals with a random monotone non-negative set function in a separable metric space X. If X is complete and if is subadditive and right continuous¹ in probability on certain classes of sets, we show the existence of a version of with right-continuous sample functions. If X is locally compact and is left continuous in probability on a certain class of open sets, there is a left-continuous version. With appropriate additional assumptions, we obtain versions that are measures or capacities. In the latter case, a 0–1 valued set function represents a random closed or compact set. The form of integer-valued strongly subadditive set functions is described for certain cases.Supported in part by National Science Foundation Grant GP-6216