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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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  • [1] J. R. Choksi, Inverse limits of measure spaces, Proc. London Math. Soc. (3) 8 (1958), 321-342. MR 0096768 (20:3251)
  • [2] E. O. Elliott and A. P. Morse, General product measures, Trans. Amer. Math. Soc. (2) 110 (1964), 245-283. MR 0158957 (28:2178)
  • [3] E. O. Elliott, Measures on countable product spaces, Pacific J. Math. (to appear). MR 0249563 (40:2808)

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Article copyright: © Copyright 1968 American Mathematical Society

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