Limits of inverse systems of measures

Inverse (or projective) systems of measure spaces (see definition 1.1. Sec. I) are used in many areas of mathematics, for example in problems connected with stochastic processes, martingales, etc. One of the first (implicit) uses was made by Kolmogoroff [9], to obtain probabilities on infinite Cartesian product spaces. The concept was later studied explicitly by Bochner, who called such systems stochastic families (see [3]). Since then, inverse systems of measure spaces have been the subject of a number of inbestigations (see e.g. Choksi [5], Metivier [12], Meyer [13], Raoult [16], Scheffer [17], [18]). The fundamental problem in all of these investigations is that of finding a "limit" for an inverse system of measure spaces (X , p , JLI , I). All previous workers in this field have concentrated on getting an appropriate 'limit' measure on the inverse limit set L. Such an approach presents some serious difficulties, e.g. L may be empty. In this paper, we avoid dependence on L, and hence many of these difficulties, by constructing on the Cartesian product X of the X,'s, a 'limit' measure TJL which retains essential features of the usual limit measure. As a result, we are able to get existence theorems with considerably fewer conditions on the systems, (in a recent paper [18], C.L. Scheffer also gets away from dependence on L by working on an abstract representation space. His methods, however, seem to be very different from ours).