Nonabsolutely convergent integrals (Q1092184)

The authors present various extensions on nonabsolutely convergent integrals given by \textit{H. W. Ellis} [Can. J. Math. 3, 471--485 (1951; Zbl 0044.05501)], \textit{J. Foran} [Proc. Am. Math. Soc. 49, 359--365 (1975; Zbl 0305.26004)] and \textit{C. M. Lee} [Fundam. Math. 100, 69--74 (1978; Zbl 0384.26004)]. The main tool is the following uniqueness theorem: A Darboux Baire 1 function \(f\) with a.e. level set at most countable is absolutely continuous on an interval \(I\) iff the following two conditions are satisfied: 1. \(f\) is absolutely continuous on every closed subset \(F\subset I\) on which it has bounded variation, 2. \(\int_{P}f'(x)\,dx<\infty,\) where \(P=\{x: 0\leq f'(x)<\infty \}\).