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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An extension theorem for obtaining measures on uncountable product spaces


Author: E. O. Elliott
Journal: Proc. Amer. Math. Soc. 19 (1968), 1089-1093
MSC: Primary 28.40
MathSciNet review: 0240271
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1968-0240271-X
PII: S 0002-9939(1968)0240271-X
Article copyright: © Copyright 1968 American Mathematical Society