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What’s the price of a nonmeasurable set? (English) Zbl 1006.28003
Summary: 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.
##### MSC:
 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28E15 Other connections with logic and set theory
##### Keywords:
Lebesgue measure; nonmeasurable set; axiom of choice
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