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Article

Keywords:
variational measures and derivates of set functions; Riemann generalized integrals
Summary:
We study properties of variational measures associated with certain conditionally convergent integrals in ${\mathbb R}^m$. In particular we give a full descriptive characterization of these integrals.
References:
[1] B. Bongiorno: Essential variation. Measure Theory Oberwolfach 1981. Lecture Notes in Math. No. 945, Springer-Verlag, Berlin, 1981, pp. 187–193. MR 0675282
[BDP] B. Bongiorno, L. Di Piazza and D. Preiss: Infinite variation and derivates in ${R}^m$. J. Math. Anal. Appl. 224 (1998), 22–33. MR 1632942
[3] B. Bongiorno, L. Di Piazza and V. Skvortsov: The essential variation of a function and some convergence theorems. Anal. Math. 22 (1996), 3–12. MR 1384345
[BDS] B. Bongiorno, L. Di Piazza and V. Skvortsov: A new full descriptive characterization of Denjoy-Perron integral. Real Anal. Exchange 21 (1995–96), 656–663. MR 1407278
[2] Z. Buczolich and W. F. Pfeffer: On absolute continuity. J. Math. Anal. Appl. 222 (1998), 64–78. MR 1623859
[4] J. Jarník and J. Kurzweil: Perron-type integration on $n$-dimensional intervals and its properties. Czechoslovak Math. J. 45 (120) (1995), 79–106. MR 1314532
[5] J. Jarník and J. Kurzweil: Differentiability and integrability in $n$-dimension with respect to $\alpha $-regular intervals. Results Math. 21 (1992), 138–151. MR 1146639
[6] J. Mawhin: Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields. Czechoslovak Math. J. 31 (106) (1981), 614–632. MR 0631606 | Zbl 0562.26004
[Mc] E. J. McShane: Unified integration. Academic Press, New York, 1983. MR 0740710 | Zbl 0551.28001
[O] K. M. Ostaszewski: Henstock integration in the plane. Mem. Amer. Math. Soc. 253 (1986). MR 0856159 | Zbl 0596.26005
[7] W. F. Pfeffer: The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge, 1993. MR 1268404 | Zbl 0804.26005
[P] W. F. Pfeffer: On variation of functions of one real variable. Comment. Math. Univ. Carolin. 38 (1997), 61–71. MR 1455470
[8] W. F. Pfeffer: On additive continuous functions of figures. Rend. Instit. Mat. Univ. Trieste, suppl. 29 (1998), 115–133. MR 1696024 | Zbl 0921.26008
[9] W. Rudin: Real and complex analysis. McGraw-Hill, New York, 1987. MR 0924157 | Zbl 0925.00005
[10] S. Saks: Theory of the integral. Dover, New York, 1964. MR 0167578
[11] B. S. Thomson: Derivates of intervals functions. Mem. Amer. Math. Soc. 452 (1991). MR 1078198
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