Let R, μ and Mμ denote the set of real numbers, Lebesgue outer measure and the class of Lebesgue measurable subsets of R respectively. It is easy to prove that the complement Ec of E ∈ Mμ is a set of Lebesgue measure zero if the inequality holds for some δ > 0 and all intervals I of R. However, in Hewitt [1], raised a problem whether the result is still true if E is not a priori measurable set. In this paper, a negative answer to this question is given through a counter-example. Also, it is proved that for a given set E ∈ M,μ with μ(E) > 0 there is a non-measurable subset A of E satisfying μ(E)