On the Sierpiński-Erdős and the Oxtoby-Ulam theorems for some new sigma-ideals of sets
HTML articles powered by AMS MathViewer
- by C. G. Mendez PDF
- Proc. Amer. Math. Soc. 72 (1978), 182-188 Request permission
Abstract:
Let $\Phi (\Psi )$ denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. There is a homeomorphism of the unit square onto itself mapping a given set in $\Phi (\Psi )$) onto a set of Lebesgue measure zero. Theorem 2. There is a set belonging to both $\Phi$ and $\Psi$ that cannot be mapped onto a set of first category by a homeomorphism of the unit square onto itself. Let C denote the Cantor set, regarded as the product of a sequence of 2-element groups, and let $\Lambda$ denote one of the $\sigma$-ideals of subsets of C studied by Schmidt and Mycielski. Theorem 3. Assuming the continuum hypothesis, the Sierpiński-Erdös theorem holds for $\Lambda$ and the class of subsets of C of Haar measure zero (or of first category). Theorem 4. The Oxtoby-Ulam theorem holds for the image of $\Lambda$ under the Cantor mapping of C onto the unit interval.References
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- C. G. Mendez, On sigma-ideals of sets, Proc. Amer. Math. Soc. 60 (1976), 124–128. MR 417359, DOI 10.1090/S0002-9939-1976-0417359-8
- Jan Mycielski, Some new ideals of sets on the real line, Colloq. Math. 20 (1969), 71–76. MR 241595, DOI 10.4064/cm-20-1-71-76
- John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443, DOI 10.1007/978-1-4684-9339-9 J. C. Oxtoby and S. M. Ulam, On the equivalence of any set of first category to a set o measure zero, Fund. Math. 31 (1938), 201-206.
- J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 5803, DOI 10.2307/1968772
- Wolfgang M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178–199. MR 195595, DOI 10.1090/S0002-9947-1966-0195595-4 E. Szpilrajn, The characteristic function of a sequence of sets and some of its applications, Fund. Math. 31 (1938), 207-225.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 72 (1978), 182-188
- MSC: Primary 54H05; Secondary 28A65, 90A05
- DOI: https://doi.org/10.1090/S0002-9939-1978-0515115-5
- MathSciNet review: 0515115