A Riesz-type definition of the Denjoy integral
A sequence \((f_ n)\) is said to be control-convergent to f on [a,b] iff \((i)\quad f_ n(x)\to f(x)\) a.e. in [a,b] as \(n\to \infty\) and each \(f_ n\) is Denjoy integrable on [a,b]; (ii) the primitives \(F_ n\) of \(f_ n\) are ACG\({}_*\) uniformly in n; (iii) \(F_ n(x)\) converges uniformly on [a,b], and \(n\to \infty.\) If \((f_ n)\) is control-convergent to f on [a,b], then f is Denjoy integrable on [a,b] and \[ \int^{b}_{a}f(x)dx=\lim_{n\to \infty}\int^{b}_{a}f_ n(x)dx. \] A function f is said to be RD integrable on [a,b] iff it is an a.e. limit of a control-convergent sequence of step functions \(\phi_ n\) and \[ \int^{b}_{a}f(x)dx=\lim_{n\to \infty}\int^{b}_{a}\phi_ n(x)dx. \] A function f is RD integrable on [a,b] if and only if it is Denjoy integrable on [a,b].
scientific article; zbMATH DE number 3975262