What’s the price of a nonmeasurable set?

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.