Equivalence of Totally Finite Measures on Infinite Product Spaces

Abstract

The object of this investigation is to study the problem of equivalence of totally finite measures on countably infinite product spaces. The Annals of Mathematics of 1948 contained a paper by S. Kakutani [4] giving necessary and sufficient conditions for equivalence of direct product measures on such spaces. These conditions were formulated in terms of a quasi-metric on the space of all such measures. Shortly thereafter, a paper by Y. Kawada [5] appeared in Mathematica Japonicae, which, among other things, gives a theorem on equivalence of measures, without the restriction to direct product measures. This theorem is stated in terms of Radon-Nikodym density functions rather than in terms of a metric. In his paper, Kawada makes use of the idea of the projection of a measure. In doing so, he is able to avoid certain difficulties involved in the usual method of defining measures on product spaces ([1], and [3] section 49). This idea is adopted in the present work for the same reason. H. D. Brunk [2] has extended the theorem of Kakutani and applied it to certain problems concerning limit theorems of probability. In his paper, Brunk deals with an interesting class of sets which had been considered earlier by C. Visser [7]. We shall define this class later and designate it throughout this report by F. In the course of his investigation, Brunk raised questions that led to the formulation of Theorem 5.A of this paper. This theorem, proved by the author, states necessary and sufficient conditions for equivalence of infinite direct product measures in terms involving the class F. It appears in this paper as a special case of the more general Theorem 4.A, which is not restricted to direct product measures. The equivalence theorems developed in this investigation relate certain subequivalence conditions on suitable subclasses of the measurable sets to the condition of equivalence, which by definition is a condition on the whole class of measurable sets. There are found in the literature two concepts of absolute continuity (and hence of equivalence) which under the finiteness condition assumed for the product measures imply one another. But, when these conditions are applied to subclasses of the measureable sets, the essential difference of these concepts becomes important. Terminology and notation is introduced to indicate these relations on subclasses of the class of measurable sets. In these terms, sufficient conditions for equivalence of measures are derived. The results of the study of these sub-equivalence conditions are combined