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aequationes mathematicae

, Volume 30, Issue 1, pp 281–283 | Cite as

A short zero-one law proof of a result of Abian

  • Harry I. Miller
Research Papers

Summary

In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero.

Let {a, b, c, ...} be a Hamel basis for the real numbers. LetA be the subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use the basis elementa. Sierpinski [4, p. 108] has shown thatA is nonmeasurable in the sense of Lebesgue. Abian [1] has improved Sierpinski's result by showing thatm* (A), the outer measure ofA, is one and thatm* (A), the inner measure ofA, is zero. In this note a very short proof, using a zero-one law, of Abian's result will be presented.

The following zero-one law is an immediate consequence of the Lebesgue Density Theorem [2, p. 290].

AMS (1980) subject classification

Primary 28A05 

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References

  1. [1]
    Abian, A.,The outer and inner measure of a nonmeasurable set. Boll. Un. Mat. Ital. (4)3 (1970), 555–558.Google Scholar
  2. [2]
    Munroe, M. E.,Introduction to measure and integration. Addison-Wesley, Cambridge, Mass., 1953.Google Scholar
  3. [3]
    Oxtoby, J. C.,Measure and category. A survey of the analogies between topological and measure spaces. Graduate Texts in Mathematics, Vol. 2. Springer, New York-Berlin, 1971.Google Scholar
  4. [4]
    Sierpinski, W.,Sur la question de la mesurabilité de la base de M. Hamel. Fund. Math.1 (1920), 105–111.Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • Harry I. Miller
    • 1
  1. 1.Department of MathematicsUniversity of SarajevoSarajevoYugoslavia

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