For a given commutative group $G$ equipped with a nonzero $\sigma$-finite diffused measure $\mu$, the question of the existence of a subgroup of $G$, nonmeasurable with respect to $\mu$, is discussed.
References
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A. B. Kharazishvili, Transformation Groups and Invariant Measures, World Scientific Publishing Co., Singapore, 1998. MR1692839 1013.28012 A. B. Kharazishvili, Transformation Groups and Invariant Measures, World Scientific Publishing Co., Singapore, 1998. MR1692839 1013.28012