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...