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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

Citations:

Zbl 0921.26006
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