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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A solution of Ulam’s problem on relative measure
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by Tim Carlson PDF
Proc. Amer. Math. Soc. 94 (1985), 129-134 Request permission

Abstract:

Suppose $\mathcal {A}$ is a collection of subsets of the unit interval and, for $A \in \mathcal {A}$, ${\mu _A}$ is a Borel measure on $A$ which vanishes on points and gives $A$ measure 1. The system ${\mu _A}(A \in \mathcal {A})$ is called a coherent system if ${\mu _A}(C) = {\mu _A}(B){\mu _B}(C)$ whenever $A$ $B \supseteq C$ are in $\mathcal {A}$ and all terms are defined. The existence of a coherent system for the collection of perfect sets is shown to be independent of Zermelo-Fraenkel set theory with the axiom of dependent choices.
References
    F. Hausdorff, Summen von ${\aleph _1}$ Mengen, Fund. Math. 26 (1934), 241-255.
  • Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
  • Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
  • S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. MR 0280310
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 94 (1985), 129-134
  • MSC: Primary 03E25; Secondary 03E75, 28A05
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0781070-X
  • MathSciNet review: 781070