On Haar measurable additive maps on \({\mathfrak P}({\mathbb{N}})\) (Q1820262)

In 1971 \textit{J. P. R. Christensen} [Math. Scand. 29(1971), 245-255 (1972; Zbl 0234.54024); Papers ''Open House Probab.'' Mat. Inst. Aarhus Univ. 1971; various Publ. Ser. 21, 32-46 (1972; Zbl 0245.28001)] proved the following beautiful result: Let \(\nu\) be a real-valued, finitely-additive set-function defined on the \(\sigma\)-algebra of all subsets of \({\mathbb{N}}\). If, when viewed as a map of the compact space \(D:=\{0,1\}^{{\mathbb{N}}}\) into \({\mathbb{R}}\), \(\nu\) is Borel measurable, then \(\nu\) is countably additive. En route he proved that if \(\nu\) is only measurable with respect to the \(\sigma\)-algebra of subsets of D which have the Baire property and if \(\nu\) annihilates all finite sets, then \(\nu =0.\) These results remain valid if the rangespace \({\mathbb{R}}\) is allowed to be any Abelian, Hausdorff topological group G, as was proved a little later by \textit{N. J. M. Andersen} and \textit{J. P. R. Christensen} [Isr. J. Math. 15, 414-420 (1973; Zbl 0291.22005)]. The author of the present paper takes up this theme. Let \(\mu\) denote Haar measure in D. A G-valued function f on D is called Lusin-\(\mu\)-measurable if the conclusion of Lusin's theorem holds for it: given \(\epsilon >0,\) there exists a compact subset K of D, such that \(\mu (D\setminus K)<\epsilon\) and the restriction of f to K is a continuous map of K into G. The author proves the equivalence of the following: (i) \(\nu\) is constant \(\mu\)-a.e. (ii) \(\nu\) is Lusin-\(\mu\)-measurable and annihilates every finite subset of \({\mathbb{N}}\), (iii) \(\nu\) is Lusin-\(\mu\)- measurable and atomless. He also considers the vector lattice of exhaustive, additive, real-valued set functions on the power set of \({\mathbb{N}}\) and shows that those which are \(\mu\)-measurable form a band. The paper is concluded with examples which illustrate the sharpness of these results and resolve negatively some conjectures which naturally suggest themselves in this set-up. An incidental benefit is a new proof of a 1917 result of \textit{W. Sierpiński} and \textit{N. Lusin} [C. R. Acad. Sci., Paris 165, 422-424 (1917)]: Let \(\lambda\) denote Lebesgue measure on \([0,1],\lambda^*\), \(\lambda_*\) the corresponding outer and inner measures; then there exists a partition \(\{A_{\alpha}\},\quad \alpha \in {\mathbb{R}},\) of \([0,1]\) such that \(\lambda^*(A_{\alpha})=1\) and \(\lambda_*(A_{\alpha})=0\) for every \(\alpha\).