What’s the price of a nonmeasurable set?

Mirko Sardella; Guido Ziliotti

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 1, page 41-48
  • ISSN: 0862-7959

Abstract

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In this note, we prove that the countable compactness of { 0 , 1 } ℝ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of ℝ . This is done by providing a family of nonmeasurable subsets of ℝ whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom of Choice, the third is devoted to the proofs.

How to cite

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Sardella, Mirko, and Ziliotti, Guido. "What’s the price of a nonmeasurable set?." Mathematica Bohemica 127.1 (2002): 41-48. <http://eudml.org/doc/249032>.

@article{Sardella2002,
abstract = {In this note, we prove that the countable compactness of $\lbrace 0,1\rbrace ^\{\{\mathbb \{R\}\}\}$ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of $\{\mathbb \{R\}\}$. This is done by providing a family of nonmeasurable subsets of $\{\mathbb \{R\}\}$ whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom of Choice, the third is devoted to the proofs.},
author = {Sardella, Mirko, Ziliotti, Guido},
journal = {Mathematica Bohemica},
keywords = {Lebesgue measure; nonmeasurable set; axiom of choice; Lebesgue measure; nonmeasurable set; axiom of choice},
language = {eng},
number = {1},
pages = {41-48},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {What’s the price of a nonmeasurable set?},
url = {http://eudml.org/doc/249032},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Sardella, Mirko
AU - Ziliotti, Guido
TI - What’s the price of a nonmeasurable set?
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 41
EP - 48
AB - In this note, we prove that the countable compactness of $\lbrace 0,1\rbrace ^{{\mathbb {R}}}$ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of ${\mathbb {R}}$. This is done by providing a family of nonmeasurable subsets of ${\mathbb {R}}$ whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom of Choice, the third is devoted to the proofs.
LA - eng
KW - Lebesgue measure; nonmeasurable set; axiom of choice; Lebesgue measure; nonmeasurable set; axiom of choice
UR - http://eudml.org/doc/249032
ER -

References

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