There are measurable Hamel functions (Q416444)

The starting point of this paper is the notion of a Hamel basis in the set \(\mathbb R^2\), where \(\mathbb R\) is the set of reals. Recently, \textit{K. Płotka} [Proc. Am. Math. Soc. 131, No. 4, 1031--1041 (2003; Zbl 1012.15001)] introduced and studied the notion of a Hamel function, namely a function such that the set \(H = \{(x, f (x))\mid x\in\mathbb R \}\) is a Hamel basis of \(\mathbb R^2\).NEWLINENEWLINE In this paper the authors investigate properties of the class of Hamel functions and obtain further results. They prove that there are Hamel functions which are measurable with respect to some \(\sigma\)-fields.NEWLINENEWLINE More precisely, they prove that there exists a Hamel function which is measurable with respect to the \(\sigma\)-field of all Marczewski measurable subsets of \(\mathbb R\) (Theorem 1), and there exists a Hamel function which is measurable with respect to the \(\sigma\)-field \(\mathrm{Bor}\Delta \mathcal I\) (Theorem 2), where \(\mathcal I\) is a \(\sigma\)-ideal of subsets or containing singletons, and Bor is the \(\sigma\)-field of Borel subsets of \(\mathbb R\).NEWLINENEWLINE As an application of the above work they examine the case, where \(\mathcal I\) is some special \(\sigma\)-ideal, namely:NEWLINENEWLINE-- the \(\sigma\)-ideal generated by the closed Lebesgue null sets,NEWLINENEWLINE-- the \(\sigma\)-ideal generated by the porous sets, orNEWLINENEWLINE-- the \(\sigma\)-ideal of sets which are both Lebesgue null and of the first category,NEWLINENEWLINE and obtain some specialized results (Corollary 6).NEWLINENEWLINEThis is an interesting contribution to function space theory.