A partition of an uncountable solvable group into three negligible subsets (Q2786929)

A measure on a \(\sigma\)-algebra of subsets of a group \(G\) is called \(G\)-invariant if it is invariant with respect to any left translation. A subset of \(X\subseteq G\) is called \(G\)-negligible if (a) there exists at least one nonzero \(\sigma\)-finite \(G\)-invariant measure \(\mu\) on \(G\) such that \(X\in \text{dom}(\mu)\), and (b) for every \(\sigma\)-finite \(G\)-invariant measure \(\nu\) on \(G\) with \(X\in \text{dom}(\nu)\) the equality \(\nu(X)=0\) holds true. The main result says: Every uncountable solvable group \(G\) admits a partition into three \(G\)-negligible sets.