##
**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)

### MSC:

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 |

### Keywords:

Riemann type integrals; additive functions; variational measures; derivatives; bounded sets of finite perimeter
PDF
BibTeX
XML
Cite

\textit{Z. Buczolich} and \textit{W. F. Pfeffer}, Czech. Math. J. 47, No. 3, 525--555 (1997; Zbl 0903.26004)

### References:

[1] | B. Bongiorno: Essential variation. Measure Theory Oberwolfach 1981, Springer Lecture Notes Math., 945, 1981, pp. 187-193. |

[2] | B. Bongiorno, L. Di Piazza and V. Skvortsov: The essential variation of a function and some convergence theorems. Anal. Math. 22 (1996), no. 1, 3-12. · Zbl 0864.26006 |

[3] | B. Bongiorno, W.F. Pfeffer and B.S. Thomson: A full descriptive definition of the gage integral. Canadian Math. Bull. 39 (1996), no. 4, 390-401. · Zbl 0885.26008 |

[4] | B. Bongiorno and P. Vetro: Su un teorema di F. Riesz. Atti Acc. Sci. Lettere e Arti Palermo (IV) 37 (1979), 3-13. |

[5] | L. Di Piazza: A note on additive functions of intervals. Real Anal. Ex. 20(2) (1994-95), 815-818. |

[6] | L.C. Evans and R.F. Gariepy: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992. · Zbl 0804.28001 |

[7] | H. Federer: Geometric Measure Theory. Springer-Verlag, New York, 1969. · Zbl 0176.00801 |

[8] | E. Giusti: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel, 1984. · Zbl 0545.49018 |

[9] | W.F. Pfeffer: The Gauss-Green theorem. Adv. Math. 87 (1991), 93-147. · Zbl 0732.26013 |

[10] | W.F. Pfeffer: A descriptive definition of a variational integral and applications. Indiana Univ. Math. J. 40 (1991), 259-270. · Zbl 0747.26010 |

[11] | W.F. Pfeffer: The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge, 1993. · Zbl 0804.26005 |

[12] | S. Saks: Theory of the Integral. Dover, New York, 1964. · Zbl 1196.28001 |

[13] | E.M. Stein: Singular Integrals and Differentiability Properties of Function. Princeton Univ. Press, Princeton, 1970. · Zbl 0207.13501 |

[14] | B. S. Thomson: Derivates of Interval Functions. Mem. Amer. Math. Soc., #452, Providence, 1991. · Zbl 0734.26003 |

[15] | A.I. Volpert: The spaces \(BV\) and quasilinear equations. Math. USSR-SB. 2 (1967), 255-267. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.