Absolutely negligible sets in uncountable groups are considered in connection with the measure extension problem (for $\sigma$-finite invariant or quasi-invariant measures). In particular, it is proved that, for any uncountable solvable group $(G,\cdot)$, there exists a countable covering
of $G$ consisting of $G$-absolutely negligible sets.
Solvable group, invariant measure, quasi-invariant measure, absolutely negligible set, Sierpi\'nski's problem.
MSC 2000: 28A05, 28D05