A combinatorial problem on translation-invariant extensions of the Lebesgue measure

It is proved that, for every natural number k≥2, there exist k subsets of the real line such that any k−1 of them can be made measurable with respect to a translation-invariant extension of the Lebesgue measure, but there is no nonzero σ-finite translation-quasi-invariant measure for which all of these k subsets become measurable. In connection with this result, a related open problem is posed.