We introduce an axiomatic approach to the theory of non-absolutely convergent integrals. The definition of our ν-integral will be descriptive and depends mainly on characteristic null conditions. By specializing our concepts we will later obtain concrete theories of integration with natural properties and very general versions of the divergence theorem.
@article{bwmeta1.element.bwnjournal-article-fmv145i3p221bwm,
author = {W. Jurkat and D. Nonnenmacher},
title = {An axiomatic theory of non-absolutely convergent integrals in Rn},
journal = {Fundamenta Mathematicae},
volume = {144},
year = {1994},
pages = {221-242},
zbl = {0824.26007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv145i3p221bwm}
}
Jurkat, W.; Nonnenmacher, D. An axiomatic theory of non-absolutely convergent integrals in Rn. Fundamenta Mathematicae, Tome 144 (1994) pp. 221-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv145i3p221bwm/
[00000] [Fed] H. Federer, Geometric Measure Theory, Springer, New York, 1969.
[00001] [Jar-Ku 1] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits -transformations, Časopis Pěst. Mat. 109 (1984), 157-167. | Zbl 0555.26005
[00002] [Jar-Ku 2] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (110) (1985), 116-139. | Zbl 0614.26007
[00003] [Jar-Ku 3] J. Jarník and J. Kurzweil, A new and more powerful concept of the PU-integral, ibid. 38 (113) (1988), 8-48. | Zbl 0669.26006
[00004] [Ju] W. B. Jurkat, The Divergence Theorem and Perron integration with exceptional sets, ibid. 43 (118) (1993), 27-45.
[00005] [Ju-Kn] W. B. Jurkat and R. W. Knizia, A characterization of multi-dimensional Perron integrals and the fundamental theorem, Canad. J. Math. 43 (1991), 526-539. | Zbl 0733.26008
[00006] [Kir] M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108. | Zbl 60.0532.03
[00007] [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31 (106) (1981), 614-632. | Zbl 0562.26004
[00008] [McSh] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842. | Zbl 0010.34606
[00009] [No] D. J. F. Nonnenmacher, Theorie mehrdimensionaler Perron-Integrale mit Ausnahmemengen, PhD thesis, Univ. of Ulm, 1990. | Zbl 0724.26010
[00010] [Pf 1] W. F. Pfeffer, The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc. 43 (1987), 143-170. | Zbl 0638.26011
[00011] [Pf 2] W. F. Pfeffer, The Gauß-Green Theorem, Adv. in Math. 87 (1991), 93-147. | Zbl 0732.26013
[00012] [Pf 3] W. F. Pfeffer, A descriptive definition of a variational integral and applications, Indiana Univ. Math. J. 40 (1991), 259-270. | Zbl 0747.26010
[00013] [Pf-Ya] W. F. Pfeffer and W.-C. Yang, A multidimensional variational integral and its extensions, Real Anal. Exchange 15 (1989-1990), 111-169.
[00014] [Rot] J. J. Rotman, An Introduction to Algebraic Topology, Graduate Texts in Math., Springer, 1988.
[00015] [Saks] S. Saks, Theory of the Integral, Dover, New York, 1964.
[00016] [Weir] A. J. Weir, General Integration and Measure, Vol. 2, Cambridge University Press, 1974. | Zbl 0286.28007
[00017] [Yee-Na] L. P. Yee and W. Naak-In, A direct proof that Henstock and Denjoy integrals are equivalent, Bull. Malaysian Math. Soc (2) 5 (1982), 43-47. | Zbl 0501.26006