Completely random measures

The theory of stochastic processes is concerned with random functions defined on some parameter set. This paper is concerned with the case, which occurs naturally in some practical situations, in which the parameter set is a ^-algebra of subsets of some space, and the random functions are all measures on this space. Among all such random measures are distinguished some which are called completely random, which have the property that the values they take on disjoint subsets are independent. A representation theorem is proved for all completely random measures satisfying a weak finiteness condition, and as a consequence it is shown that all such measures are necessarily purely atomic.