A new non-measurable set as a vector space
Commun. Korean Math. Soc. 2006 Vol. 21, No. 3, 429-432
Printed September 1, 2006
Soon-Yeong Chung
Sogang University
Abstract : We use Cauchy's functional equation to construct a new non-measurable set which is a (vector) subspace of $\Bbb R$ and is of a codimension 1, considering $\Bbb R$, the set of real numbers, as a vector space over a field $\Bbb Q$ of rational numbers. Moreover, we show that ${\Bbb R}$ can be partitioned into a countable family of disjoint non-measurable subsets.
Keywords : non-measurable set
MSC numbers : 28A05
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