On absolutely Henstock integrable functions (Q5903803)

A non-negative function f is said to be RL-integrable on [a,b] if there exists a number A such that for every \(\epsilon >0\) there exist an open set G and a constant \(\delta >0\) such that \(| G| <\epsilon\) and that for every division \(a=x_ 0<x_ 1<...<x_ n=b\) with \(x_ i- x_{i-1}<\delta\) for \(i=1,2,...,n\) and for any \(\xi_ i\in [x_{i- 1},x_ i]\setminus G\) we have \[ | \sum^{n}_{i=1}f(\xi_ i)(x_ i-x_{i-1})-A| <\epsilon. \] A function f is RL-integrable on [a,b] if both \(f^+:=\max (f,0)\) and \(f^-:=\max (-f,0)\) are RL- integrable on [a,b]. The authors prove that a function f is RL-integrable on [a,b] if and only if it is absolutely Henstock integrable on [a,b].