Paradoxical characterization of Lebesgue nonmeasurable sets

Abstract

We show that a set $A\subseteq {\mathbb{R}}^n$ is nonmeasurable in the sense of Lebesgue if and only if it has a common density point with its complement $A^c$ . Moreover, if there exists a density point of both A and $A^c$ , then the set of such points has a positive Lebesgue measure.

Abstract

Mathematics; Lebesgue measurability; Density point; Nonmeasurable set