On the Sierpiński-Erdős and the Oxtoby-Ulam theorems for some new sigma-ideals of sets

Author:
C. G. Mendez

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

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Abstract: Let 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* ) *onto a set of Lebesgue measure zero*. Theorem 2. *There is a set belonging to both* *and* *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 denote one of the -ideals of subsets of *C* studied by Schmidt and Mycielski. Theorem 3. *Assuming the continuum hypothesis, the Sierpiński-Erdös theorem holds for* *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* *under the Cantor mapping of C onto the unit interval*.

**[1]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869****[2]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[3]**C. G. Mendez,*On sigma-ideals of sets*, Proc. Amer. Math. Soc.**60**(1976), 124–128. MR**0417359**, https://doi.org/10.1090/S0002-9939-1976-0417359-8**[4]**Jan Mycielski,*Some new ideals of sets on the real line*, Colloq. Math.**20**(1969), 71–76. MR**0241595****[5]**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****[6]**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.**[7]**J. C. Oxtoby and S. M. Ulam,*Measure-preserving homeomorphisms and metrical transitivity*, Ann. of Math. (2)**42**(1941), 874–920. MR**0005803**, https://doi.org/10.2307/1968772**[8]**Wolfgang M. Schmidt,*On badly approximable numbers and certain games*, Trans. Amer. Math. Soc.**123**(1966), 178–199. MR**0195595**, https://doi.org/10.1090/S0002-9947-1966-0195595-4**[9]**E. Szpilrajn,*The characteristic function of a sequence of sets and some of its applications*, Fund. Math.**31**(1938), 207-225.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0515115-5

Keywords:
Measure zero,
first category,
ideals of sets,
infinite games,
winning strategies,
homeomorphisms,
Sierpiński-Erdös theorem,
Oxtoby-Ulam theorem

Article copyright:
© Copyright 1978
American Mathematical Society