Vector spaces of non-measurable functions
Let \((\Omega,\Sigma)\) be a measurable space and let \(\gamma\) be an infinite cardinal. Suppose there is a family \({\mathcal V}\) of pairwise disjoint subsets of \(\Omega\) with card \({\mathcal V}=\gamma\) such that the union of each nonempty subfamily of \({\mathcal V}\) is not in \(\Sigma\). Then for every Banach space \(X\) of density character \(\gamma\) there exists an injective continuous linear operator \(T: X\to\mathbb{K}^\Omega\), where \(\mathbb{K}\) is the scalar field of \(X\), with \(T(x)\) not \(\Sigma\)-measurable for each \(x\neq 0\) (Proposition 2.4). An isometric version of this result is also established. The paper is written carelessly. In particular, the crucial disjointness condition is missing in the formulations of Lemma 2.1, Theorem 2.2 and Propositions 2.3 and 2.4. On the other hand, the non-measurability assumption in Proposition 2.3 is irrelevant. Also, in the course of the proof of Proposition 2.4 the authors establish a result which is an obvious consequence of a standard proposition formulated at the end of the proof of Theorem 2.5. Finally, we note that the result established after the proof of Lemma 2.1 is well known. For example, according to a classical theorem due to \textit{N. Lusin} and \textit{W. Sierpiński} [C.R. 165, 422-424 (1917; JFM 46.0294.01)] the unit interval can be partitioned into \({\mathfrak c}\)-many sets of Lebesgue outer measure~1.
scientific article; zbMATH DE number 5156877
scientific article