A solution of Ulam's problem on relative measure
Proc. Amer. Math. Soc. 94 (1985), 129-134
Primary 03E25; Secondary 03E75, 28A05
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Abstract: Suppose is a collection of subsets of the unit interval and, for , is a Borel measure on which vanishes on points and gives measure 1. The system is called a coherent system if whenever are in 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.
F. Hausdorff, Summen von Mengen, Fund. Math. 26 (1934), 241-255.
Jech, Set theory, Academic Press [Harcourt Brace Jovanovich
Publishers], New York, 1978. Pure and Applied Mathematics. MR 506523
M. Solovay, A model of set-theory in which every set of reals is
Lebesgue measurable, Ann. of Math. (2) 92 (1970),
0265151 (42 #64)
M. Ulam, Problems in modern mathematics, Science Editions John
Wiley & Sons, Inc., New York, 1964. MR 0280310
- F. Hausdorff, Summen von Mengen, Fund. Math. 26 (1934), 241-255.
- T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
- R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56. MR 0265151 (42:64)
- S. M. Ulam, Problems in modern mathematics, Wiley, New York, 1964. MR 0280310 (43:6031)
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