## A fundamental theorem of calculus for the Kurzweil-Henstock integral in $$\mathbb{R}^m$$.(English)Zbl 1024.26005

Let $$f$$ be a real-valued function on a compact interval in $$\mathbb{R}^m$$ and let $$F$$ be a real-valued set function defined on intervals. Define $$\Gamma_\varepsilon=\{(x,I):|F(I)-f(x)|I||\geq\varepsilon|I|\}$$. If $$\delta$$ is a gauge on $$E$$ then let $$X(\varepsilon, \delta)=\{x\in E: \text{ there is a } \delta\text{-fine } (x,I)\in\Gamma_\varepsilon\}$$. The set $$X(\varepsilon, \delta)$$ is considered as a set of singularities that is larger than that considered by J. T. Lu and P.-Y. Lee [“The primitives of Henstock integrable functions in Euclidean space”, Bull. Lond. Math. Soc. 31, No. 2, 173-180 (1999; Zbl 0921.26006)] in their work on the differentiability of the primitive. The authors prove that $$f$$ is Henstock-Kurzweil integrable and $$F$$ is its primitive if and only if for all $$\varepsilon>0$$ there is a gauge $$\delta:E\to (0,1)$$ such that $$\sum|F(E)|<\varepsilon$$ and $$\sum|f(x)||I|<\varepsilon$$ for sums over every $$\delta$$-fine partial division in $$\Gamma_\varepsilon$$.
Also considered are various convergence theorems using a type of strong Lusin condition.

### MSC:

 26B15 Integration of real functions of several variables: length, area, volume 26A39 Denjoy and Perron integrals, other special integrals

Zbl 0921.26006