On the equivalence of Henstock-Kurzweil and restricted Denjoy integrals in \(R^ n\) (Q916807)

Let \(E\subset R^ n\) be a nondegenerated closed interval. An interval function F is said to be differentiable at a point \(x\in E\) with derivative \(F'(x)\) iff for every \(\epsilon >0\) there exist functions \[ (1)\quad \delta (x)>0,\quad 1>\rho (x)>0 \] defined in E such that whenever an interval I fulfills \[ (2)\quad x\in I,\quad diam(I)\leq \delta (x),\quad \rho (x)diam(I)^ n\leq | I| \] we have \(| F(I)-F'(x)| I| | \leq \epsilon | I|.\) Let \(X\subset E\). An interval function F is said to be \(AC^*(X)\) iff for every \(\epsilon >0\) there exist \(\eta >0\) and functions \(\delta\), \(\rho\) fulfilling (1) and such that for any partial partition \((I_ i)\) of E and for any \(x_ i\in I_ i\) if \(\sum | I_ i| <\eta\) and condition (2) holds for pairs \((I_ i,x_ i)\), then \(\sum | F(I_ i)| \leq \epsilon.\) A function f defined on E is said to be restricted Denjoy integrable on E iff there exists a continuous additive interval function F and E is the union of a sequence \((X_ i)\) for which F is \(AC^*(X_ i)\) and \(F'(x)=f(x)\) for almost all \(x\in E.\) A function f defined on E is said to be Henstock-Kurzweil integrable on \(E\subset R^ n\) if there exists a number A such that for every \(\epsilon >0\) there exist functions \(\delta\), \(\rho\) for which (1) holds and \(| f(x)| I| -A| \leq \epsilon\) whenever (I) is a partition of E with associate points \(x\in I\) such that (2) holds and \(\sum\) sums over (I). Theorem. A function f is Henstock-Kurzweil integrable on E if and only if f is restricted Denjoy integrable there.