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 σ-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.
