On the Translates of a Set which Meet it in a Set of Positive Measure

Abstract

Given a singular Borel regular measure ${m_a}$ on ${R^n}$ and a Borel subset E of ${R^n}$, it is shown that the set of vectors x for which ${m_a}((E + x) \cap E) > 0$ is of Lebesgue measure 0. This fact is then used to show that subsets of finite, nonzero, Hausdorff s-measure are nonmeasurable sets with respect to any approximating measure $s - {m_\delta }$.