ABSTRACT
The main goal of this chapter is to demonstrate that some Vitali subsets of the real line R can be measurable with respect to certain translation quasiinvariant measures on R extending the standard Lebesgue measure. So, according to the general concept introduced earlier (see Chapter 5), we may say that some Vitali sets turn out to be relatively measurable with respect to the class of all those translation quasi-invariant measures on R which extend the Lebesgue measure. On the other hand, we will also show that there exist Vitali sets which are nonmeasurable with respect to every nonzero σ-finite translation quasi-invariant measure on R. In other words, those Vitali sets turn out to be absolutely nonmeasurable with respect to the class of all nonzero σ-finite translation quasi-invariant measure on R.
