Convergence theorems for the variational integral (Q909813)

Convergence theorems for the Denjoy-Perron integral, given originally by Dzhvarshejshvili using the Denjoy theory, have recently been rediscovered and proved in various equivalent theories by Peng-Yee Lee and his co- workers [see \textit{Peng-Yee Lee}, Langzhou lectures on Henstock integration, World Scientific, Singapore (1989)]. The present author has now proved these theorems in the setting of the variational approach to the Denjoy-Perron integral due to \textit{R. Henstock} [see Lectures on the theory of integration (1988; Zbl 0668.28001)]. The basic result proved is: if \(f_ n\) is variationally integrable on \([a,b],\) with primitive \(F_ n,\quad n=1,2,...,\) and if \(f=\lim_{n\to \infty}f_ n,\quad F=\lim_{n\to \infty}F_ n,\) then f is variationally integrable on \([a,b]\) with primitive F iff: for all \(\epsilon >0\) there is an integer-valued function M, such that for infinitely many \(m\geq M\) there is a \(\delta_ m: [a,b]\to]0,\infty [,\) and a superadditive interval function S on [a,b] with \(0=S([a,a])\leq S([a,b])<\epsilon\) and if \(x-\delta_ m(x)<u\leq x\leq v<x+\delta_ m(x),\quad m=m(x),\) then \[ | F_ m(u,v)-F(u,v)| \leq S([u,v]), \] where \(F_ m(u,v)=F_ m(v)-F_ m(u)\), \(F(u,v)=F(v)-F(u)\).