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On nonmeasurable subgroups of the real line. (English) Zbl 0884.28003
If \(m\) is an arbitrary nonzero sigma finite measure on the real line which is invariant with respect to translations, then there exists a subset \(V\) of \(R\) which is a vector space over the field of rational numbers and \(V\) is nonmeasurable with respect to \(m\). The proofs of this and the remaining theorems in the paper use essentially some Hamel bases in a similar way as in a previous paper of the same author.

MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28D05 Measure-preserving transformations
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