Geodesic
What’s the price of a nonmeasurable set?
Mathematica Bohemica, Tome 127 (2002) no. 1, pp. 41-48
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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.
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.
DOI : 10.21136/MB.2002.133985
Classification : 28A05, 28A20, 28E15
Keywords: Lebesgue measure; nonmeasurable set; axiom of choice 
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Sardella, Mirko; Ziliotti, Guido. What’s the price of a nonmeasurable set?. Mathematica Bohemica, Tome 127 (2002) no. 1, pp. 41-48. doi: 10.21136/MB.2002.133985

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