MATHEMATICA BOHEMICA, Vol. 127, No. 1, pp. 41-48 (2002)

What's the price of a nonmeasurable set?

Mirko Sardella, Guido Ziliotti

Mirko Sardella, Dipartimento di Matematica, Politecnico, corso Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail:; Guido Ziliotti, Universita di Pisa, Dipartimento di Matematica, via F. Buonarroti 56100 Pisa, Italy, e-mail:

Abstract: In this note, we prove that the countable compactness of \set \^^Mtogether with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of $\re $. This is done by providing a family of nonmeasurable subsets of $\re $ 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.

Keywords: Lebesgue measure, nonmeasurable set, axiom of choice

Classification (MSC2000): 28A20, 28E15

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