What’s the price of a nonmeasurable set?

*(English)*Zbl 1006.28003Summary: We prove that the countable compactness of \(\{0,1\}^{\mathbb{R}}\) 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.