The existence of an inverse limit of an inverse system of measure spaces — A purely measurable case

Abstract

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sufficient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.

Projective Limits and Infinite Products of Probability Measures

“…7.2, Part A] for some sufficient conditions for the convergence. A sharper result has been obtained recently by Pintér [15].…”

“…The existence of projective limits of probability spaces has been extensively studied since the foundational works of Kolmogorov [11] and Bochner [1] in the fifties. The greatest efforts in previous works have been devoted to establishing necessary conditions on the projective system for the existence of the projective limit measure (see, e.g., Choksi [4], Metivier [14], Bourbaki [2], Mallory and Sion [13], Rao [16], Frolík [7], Rao and Sazonov [17] and Pintér [15], among others). In this article we will change the perspective slightly.…”

An infinitary propositional probability logic

“…Inverse limits of systems of probability spaces are known to exist only under suitable topological assumptions. See, for instance, the Introduction in [17]. In the latter work, the author introduces a purely measure-theoretic condition, called -completeness, that suffices for existence and uniqueness of inverse limits when the index set is the set of natural numbers.…”

“…Hence, -completeness of (( i , F i , T i ), ( f i j )) i≤ j,i, j∈N is equivalent to the condition that, for all ω ∈ , the inverse system of probability spaces [17,Definition 3.1].…”