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Keywords:
Lebesgue measure; nonmeasurable set; axiom of choice
Summary:
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.
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