Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A NEW NON-MEASURABLE SET AS A VECTOR SPACE

  • Chung, Soon-Yeong (Department of Mathematics and the Program of Integrated Biotechnology Sogang University)
  • Published : 2006.07.01
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Abstract

We use Cauchy's functional equation to construct a new non-measurable set which is a (vector) subspace of \mathbb{R}$ and is of a codimensiion 1, considering \mathbb{R}$, the set of real numbers, as a vector space over a field \mathbb{Q}$ of rational numbers. Moreover, we show that \mathbb{R}$ can be partitioned into a countable family of disjoint non-measurable subsets.

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References

  1. G. de Barra, Measure Theory and Integration, Ellis Horwod Limited, New York, 1981
  2. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, New York, 1952