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## Fundamenta Mathematicae

2000 | 166 | 3 | 269-279
Tytuł artykułu

### Vitali sets and Hamel bases that are Marczewski measurable

Autorzy
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Języki publikacji
EN
Abstrakty
EN
We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
269-279
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-12-24
poprawiono
2000-08-24
Twórcy
autor
• Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, U.S.A.
• Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bontchev street, bl. 8, 1113 Sofia, Bulgaria
• Department of Mathematics, Auburn University, 218 Parker Hall, Auburn, AL 36849-5310, U.S.A.
Bibliografia
• [1] C.Burstin, Die Spaltung des Kontinuums in $\cc$ in Lebesgueschem Sinne nichtmessbare Mengen, Sitzungsber. Akad. Wiss. Wien Math. Nat. Klasse Abt. IIa 125 (1916), 209-217.
• [2] P.Erdős, On some properties of Hamel bases, Colloq. Math. 10 (1963), 267-269.
• [3] J.Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras I, Trans. Amer. Math. Soc. 234 (1977), 289-324.
• [4] F. B.Jones, Measure and other properties of a Hamel basis, Bull. Amer. Math. Soc. 48 (1942), 472-481.
• [5] A. S.Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, 1995.
• [6] M.Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality, Prace Nauk. Uniw. Śląsk. 489, Uniw. Śląski, Katowice, and PWN, Warszawa, 1985.
• [7] A. W.Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), Elsevier, 1984, 201-233.
• [8] W.Sierpiński, Sur la question de la mesurabilité de la base de Hamel, Fund. Math. 1 (1920), 105-111.
• [9] J. H.Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), 1-28.
• [10] E.Szpilrajn (Marczewski), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34.
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Bibliografia
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