On necessary and sufficient conditions for non-absolute integrability (Q1898993)

\textit{Y. Kubota} [Proc. Japan Acad. 40, 713-717 (1964; Zbl 0141.246); ibid. 42, 737-742 (1966; Zbl 0145.059); ibid. 43, 441-444 (1967; Zbl 0153.382)] defined a natural extension of the general Denjoy integral, the \({\mathcal K}\)-integral, by saying that \(f\) is \({\mathcal K}\)-integrable if there exists an approximately continuous [ACG] function \(F\) with \(F_{ap}' = f\) almost everywhere; here [ACG] means that in the definition of ACG the sets in the sequence are required to be closed. The main result of the paper can be regarded as an alternative Riesz-type definition of the \({\mathcal K}\)-integral: a function \(f\) is \({\mathcal K}\)- integrable on \([a,b]\) iff there exists an increasing sequence of closed sets \(\{X_n\}^\infty_{n = 1}\) with union \([a,b]\), with \(f \in {\mathcal L} (X_n)\), \(n = 1,2, \ldots\), and if \(F_n = \int f1_{X_n}\), \(n = 1,2, \ldots, \{F_n\}^\infty_{n = 1}\) is uniformly [ACG], and \(F_n \to F\), where \(F\) is approximately continuous. A similar result is also given for the general Denjoy integral.