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Proceedings of the American Mathematical Society

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A solution of Ulam’s problem on relative measure


Author: Tim Carlson
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
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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 [Enhancements On Off] (What's this?)

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Keywords: Coherent system, forcing, perfect tree, universal measure zero
Article copyright: © Copyright 1985 American Mathematical Society