In this note, we prove that the countable compactness of ${\{0,1\}}^{ℝ}$ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of $ℝ$. This is done by providing a family of nonmeasurable subsets of $ℝ$ 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...

The paper is concerned with generalized (i. e. monotone and possibly non-additive) measures. A discussion concerning the classification of these measures, according to the type and amount of non-additivity, is done. It is proved that $\lambda$-additive measures appear naturally as solutions of functional equations generated by the idea of (possible) non additivity.