Many proofs of the fact that there exist Lebesgue nonmeasurable subsets of the real line are known. The oldest proof of this result is due to Vitali . The cosets (under addition) of Q, the set of rational numbers, constitute a partition of the line into an uncountable family of disjoint sets, each congruent to Q under translation, Vitali's proof shows that V is nonmeasurable, if V is a set having one and only one element in common with each of these cosets.
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Published by: Academy of Arts and Sciences of Bosnia and Herzegovina and Department of Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina (Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka LXIX (1982) 39-43)
Ćurgus, Branko and Miller, Harry I., "Nonmeasurable Sets and Pairs of Transfinite Sequences" (1982). Mathematics. Paper 78.