Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller; Strashimir Popvassilev

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 3, page 269-279
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Miller, Arnold, and Popvassilev, Strashimir. "Vitali sets and Hamel bases that are Marczewski measurable." Fundamenta Mathematicae 166.3 (2000): 269-279. <http://eudml.org/doc/212481>.

@article{Miller2000,
abstract = {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.},
author = {Miller, Arnold, Popvassilev, Strashimir},
journal = {Fundamenta Mathematicae},
keywords = {Marczewski measurable set; Vitali set; Hamel basis; Polish space; Marczewski null sets},
language = {eng},
number = {3},
pages = {269-279},
title = {Vitali sets and Hamel bases that are Marczewski measurable},
url = {http://eudml.org/doc/212481},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Miller, Arnold
AU - Popvassilev, Strashimir
TI - Vitali sets and Hamel bases that are Marczewski measurable
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 3
SP - 269
EP - 279
AB - 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.
LA - eng
KW - Marczewski measurable set; Vitali set; Hamel basis; Polish space; Marczewski null sets
UR - http://eudml.org/doc/212481
ER -

References

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  4. [4] F. B.Jones, Measure and other properties of a Hamel basis, Bull. Amer. Math. Soc. 48 (1942), 472-481. Zbl0063.03064
  5. [5] A. S.Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, 1995. 
  6. [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. [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. [8] W.Sierpiński, Sur la question de la mesurabilité de la base de Hamel, Fund. Math. 1 (1920), 105-111. 
  9. [9] J. H.Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), 1-28. Zbl0517.03018
  10. [10] E.Szpilrajn (Marczewski), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34. Zbl61.0229.01

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