Some equivalents of the \(AP\) controlled convergence theorem, their generalizations and a Riesz-type definition of the \(AP\)-integral (Q1898975)

The approximately continuous Perron integral was defined by \textit{J. C. Burkill} [Math. Z. 34, 270-278 (1931; Zbl 0002.38604)]. A convergence theorem known as the controlled convergence theorem for the integral has been proved [see the author, Real Anal. Exch. 19, No. 1, 212-217 (1994; Zbl 0804.26006)]. The author gives various equivalents of the controlled convergence, and, as a consequence, provides a Riesz-type definition of the integral.