Some questions concerning invariant extensions of Lebesgue measure (Q1898963)

Denote by \(M(l,\mathbb{R})\) the class of all translation invariant (\(\sigma\)-additive) measures on \(\mathbb{R}\) extending the Lebesgue measure \(l\). Using CH, the author establishes the existence of two special Sierpiński sets \(X\) and \(Y\) on \(\mathbb{R}\). The first has the property that there exists \(\mu\in M(l,\mathbb{R})\) with \(\mu(\mathbb{R}\backslash X)=0\) (Theorem 1). The second has the opposite property that there exists \(\nu\in M(l,\mathbb{R})\) such that \(Y\) does not belong to the domain of any measure in \(M(l,\mathbb{R})\) extending \(\nu\) (Theorem 2). The author calls a set \(A\subset \mathbb{R}\) negligible with respect to a class \(M\) of measures on \(\mathbb{R}\) provided that there exists \(\mu\in M\) having \(A\) in its domain and for every \(\mu\) with this property we have \(\mu(A)=0\). Using a construction due to Sierpiński, he proves the existence of a partition of \(\mathbb{R}\) into three sets negligible with respect to the class of all \(\sigma\)-finite translation invariant measures on \(\mathbb{R}\) (Theorem 5). Clearly, two sets do not suffice for this purpose. Other related results are also established. \{Reviewer's remarks: (1) Reference [2] by E. Szpilrajn (Marczewski) is now available in English translation, supplemented with comments [see \textit{E. Marczewski}: ``Collected mathematical papers'', Institute of Mathematics, Polish Academy of Sciences (1996), pp. 276-287]. (2) \textit{S. Solecki's} paper mentioned by the author in connection with Theorem 3 has appeared [Proc. Am. Math. Soc. 119, No. 1, 115-124 (1993; Zbl 0784.28006)]. (3) The author's comments on Theorem 4 are somewhat misleading. In this connection see \textit{E. Grzegorek} [Bull. Acad. Pol. Sci., Sér. Sci. Math. 28, 7-10 (1980; Zbl 0483.28003), Theorem 1]\}.