Journal of Applied Analysis Vol. 2, No. 2, pp. 171181 (1996) 

On nonmeasurable subgroups of the real lineA.B. KharazishviliInstitute of Applied MathematicsUniversity of Tbilisi University street 2 380043 Tbilisi 43 Republic of Georgia Abstract: We prove that, for every nonzero $\sigma$finite measure $\mu$ defined on the real line $R$ and invariant (or quasiinvariant) under all translations of $R$, there exists a subgroup of $R$ nonmeasurable with respect to $\mu$. Some generalizations of this result are discussed, too, and several problems related to them are posed. Keywords: Real line, invariant measure, quasiinvariant measure, nonmeasurable subgroup, Hamel basis, Ulam matrix,uncountable commutative group, Jonsson group Classification (MSC2000): 28A05, 28D05 Full text of the article:
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