Completely random measures

Abstract

The theory of stochastic processes is concerned with randomfunctions defined on some parameter set. This paper is concernedwith the case, which occurs naturally in some practicalsituations, in which the parameter set is a σ-algebra of subsetsof some space, and the random functions are all measureson this space. Among all such random measures aredistinguished some which are called completely random, whichhave the property that the values they take on disjoint subsetsare independent. A representation theorem is proved for allcompletely random measures satisfying a weak finiteness condition, and as a consequence it is shown that all such measuresare necessarily purely atomic. © 1967 by Pacific Journal of Mathematics.