An extension theorem for obtaining measures on uncountable product spaces

Abstract:

Several theorems are known for extending consistent families of measures to an inverse limit or product space [1]. In this paper the notion of a consistent family of measures is generalized so that, as with general product measures [2], the spaces are not required to be of unit measure or even $\sigma$-finite. The general extension problem may be separated into two parts, from finite to countable product spaces and from countable to uncountable product spaces. The first of these is discussed in [3]. The present paper concentrates on the second. The ultimate virtual identity of sets is defined and used as a key part of the generalization and nilsets similar to those of general product measures [2] are introduced to assure the measurability of the fundamental covering family. To exemplify the extension process, it is applied to product measures to obtain a general product measure. The paper is presented in terms of outer measures and Carathéodory measurability; however, some of the implications in terms of measure algebras should be obvious.