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Finite Borel measures on spaces of cardinality less than $ \mathfrak{c}$


Authors: R. J. Gardner and G. Gruenhage
Journal: Proc. Amer. Math. Soc. 81 (1981), 624-628
MSC: Primary 54H99; Secondary 03E15, 04A15, 28A12, 54D20, 54G20
DOI: https://doi.org/10.1090/S0002-9939-1981-0601743-5
MathSciNet review: 601743
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \kappa < c$ be uncountable. We prove, among other results, that every $ \alpha $-realcompact space of cardinality $ \kappa $ is Borel measure-compact if and only if there is a set of reals of cardinality $ \kappa $ whose Lebesgue measure is not zero.


References [Enhancements On Off] (What's this?)

  • [1] F. R. Drake, Set theory, North-Holland, London, 1974.
  • [2] N. Dykes, Generalizations of realcompact spaces, Pacific J. Math. 33 (1970), 571-581. MR 0276928 (43:2668)
  • [3] D. H. Fremlin, Uncountable powers of $ {\mathbf{R}}$ can be almost Lindelöf, Manuscripta Math. 22 (1977), 77-85. MR 0464155 (57:4090)
  • [4] R. J. Gardner, The regularity of Borel measures and Borel measure-compactness, Proc. London Math. Soc. (3) 30 (1975), 95-113. MR 0367145 (51:3387)
  • [5] A. W. Hager, G. D. Reynolds and M. D. Rice, Borel-complete topological spaces, Fund. Math. 75 (1972), 135-143. MR 0309071 (46:8182)
  • [6] P. R. Halmos, Measure theory, Van Nostrand, Princeton, N. J., 1950. MR 0033869 (11:504d)
  • [7] R. Haydon, On compactness in spaces of measures and measure-compact spaces, Proc. London Math. Soc. (3) 29 (1974), 1-16. MR 0361745 (50:14190)
  • [8] K. Kunen, Inaccessibility properties of cardinals, Doctoral Dissertation, Stanford Univ., 1968.
  • [9] D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108-111. MR 0006595 (4:12a)
  • [10] D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. MR 0270904 (42:5787)
  • [11] W. Pfeffer, Integrals and measures, Dekker, New York, 1977. MR 0460580 (57:573)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0601743-5
Keywords: Borel measure, regular measure, Borel-complete space, measure-compact space, $ \alpha $-real compact space, Martin's axiom
Article copyright: © Copyright 1981 American Mathematical Society

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