Upper and lower approximations of Perron integrable functions (Q1979041)

A characterization of Perron integrability of a function \(f:[a,b] \to \overline {\mathbb R}\) is given using upper and lower approximations of \(f\) which are lower and upper semi-Baire \(1\). For example: \(f\) is Perron integrable if and only if for every \(\varepsilon >0\) there exist two integrable functions \(g_\varepsilon \) and \(h_\varepsilon \) which are upper and lower semi-Baire \(1 \), respectively, \(g_\varepsilon < \infty \), \(h_\varepsilon >-\infty \), \(g_\varepsilon \leq f\leq h_\varepsilon \) and \(\int _a^b(h_\varepsilon - g_\varepsilon) <\varepsilon \).