A Combinatorial Problem on Translation Invariant Extensions of the Lebesgue Measure

In this chapter we are going to prove that, for every natural number k ≥ 2, there exist k many subsets of the real line R such that any k − 1 of them can be simultaneously made measurable with respect to a certain translation invariant extension of the Lebesgue measure (in general, depending on a choice of these k−1 subsets), but there is no nonzero σ-finite translation quasi-invariant measure on R for which all of these k subsets become measurable.