Journal of Applied Analysis
Vol. 2, No. 2, pp. 171-181 (1996)
On nonmeasurable subgroups of the real line
A.B. KharazishviliInstitute of Applied Mathematics
University 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
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