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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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