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Concerning non-measurable subsets of a given measurable set

  • H. W. Pu (a1)

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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 EMμ 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 EM,μ with μ(E) > 0 there is a non-measurable subset A of E satisfying μ(E)

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References

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[1]Hewitt, E. and Stromberg, K., Real and Abstract Analysis (Springer-Verlag, Berlin, 1965, p. 295).
[2]Munroe, M. E., Introduction to Measure and Integration (Addison-Wesley, Reading, Massachusetts, 1959, pp. 9697).
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