Variations of additive functions. (English) Zbl 0903.26004

Basic tools are the family \(\mathcal F\) of figures (finite unions of intervals) in \(\mathbb R^n\) and the family \(\mathcal B\mathcal V\) (essentially closed BV sets). Let \(A\in \mathcal F\) or \(A\in \mathcal B \mathcal V.\) A partition in \(A\) is a finite set of pairs \((t_i,A_i)\) where \(A_i\) is from the same family as \(A\), \(A_i\) and \(A_j\) are nonoverlapping for \(i\not = j\) and \(t_i \in A_i\). The regularity of \(A\) is measure of \(A\) divided by the product of diameter of \(A\) and perimeter of \(A\) (\((n-1)\) dimensional Hausdorff measure of the essential boundary of \(A\)). Derivatives of functions defined on \(\mathcal F\) or \(\mathcal B\mathcal V\) are introduced and a version of Ward’s theorem for additive functions of figures is proved. Two types of variations of a function \(F\) on \(\mathcal F\) (or on \(\mathcal B\mathcal V\)) on a set \(E\) are introduced which lead to Borel regular measures and to necessary and sufficient conditions for differentiability of \(F\) a.e. Finally, a conditionally convergent integral of Riemann type is defined and its descriptive characterization is given.
Reviewer: J.Kurzweil (Praha)


26B30 Absolutely continuous real functions of several variables, functions of bounded variation
28A75 Length, area, volume, other geometric measure theory
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
58C35 Integration on manifolds; measures on manifolds
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