A NEW NON-MEASURABLE SET AS A VECTOR SPACE Chung, Soon-Yeong;
We use Cauchy's functional equation to construct a new non-measurable set which is a (vector) subspace of and is of a codimensiion 1, considering , the set of real numbers, as a vector space over a field of rational numbers. Moreover, we show that can be partitioned into a countable family of disjoint non-measurable subsets.
G. de Barra, Measure Theory and Integration, Ellis Horwod Limited, New York, 1981
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, New York, 1952