An extension theorem of Baire 1 functions of several variables (Q750640)

The main result of the paper is the following theorem: If u is Baire one on \({\mathbb{R}}^ k\) and \(H\subset {\mathbb{R}}^ k\) is a set of k-dimensional Lebesgue measure zero, then there exists a function f such that each \(x\in {\mathbb{R}}^ k\) is a Lebesgue point of f and \(u(x)=f(x)\) for \(x\in H\). This is a k-dimensional version of a theorem of \textit{G. Petruska} and \textit{M. Laczkovich} [Acta Math. Sci. Hung. 25, 189-212 (1974; Zbl 0279.26003)].