On the measurability of functions of several variables (Q1086692)

Let E be a Lebesgue measurable set in \({\mathbb{R}}^{p+q}\) and Y a metric space. If \(f:| E\to Y\) is such that \(f(.,x)\) is L-measurable for almost all x and \(f(t,.)\) is continuous in each of the q variables separately for almost all t, then f must be L-measurable (Theorem 1). By this result we deduce that a function \(f:E\to Y\) is almost-continuous iff it is almost-separately continuous. Finally, we give another characterization of the measurability of a function \(f: {\mathbb{R}}^{p+q}\to Y\) by means of properties of its sections (Theorem 2).