An extension theorem for obtaining measures on uncountable product spaces
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- by E. O. Elliott
- Proc. Amer. Math. Soc. 19 (1968), 1089-1093
- DOI: https://doi.org/10.1090/S0002-9939-1968-0240271-X
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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.References
- J. R. Choksi, Inverse limits of measure spaces, Proc. London Math. Soc. (3) 8 (1958), 321–342. MR 96768, DOI 10.1112/plms/s3-8.3.321
- E. O. Elliott and A. P. Morse, General product measures, Trans. Amer. Math. Soc. 110 (1964), 245–283. MR 158957, DOI 10.1090/S0002-9947-1964-0158957-5
- E. O. Elliott, Measures on countable product spaces, Pacific J. Math. 30 (1969), 639–644. MR 249563
Bibliographic Information
- © Copyright 1968 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 19 (1968), 1089-1093
- MSC: Primary 28.40
- DOI: https://doi.org/10.1090/S0002-9939-1968-0240271-X
- MathSciNet review: 0240271