Download PDF - An axiomatic theory of non-absolutely convergent integrals in RnAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

An axiomatic theory of non-absolutely convergent integrals in Rn

Authors 1, 1

Affiliations

  1. Abteilung für Mathematik II und V, Universität Ulm, D-89069 Ulm, Germany

Abstract

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.

Bibliography

  1. [Fed] H. Federer, Geometric Measure Theory, Springer, New York, 1969.
  2. [Jar-Ku 1] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits C1-transformations, Časopis Pěst. Mat. 109 (1984), 157-167.
  3. [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.
  4. [Jar-Ku 3] J. Jarník and J. Kurzweil, A new and more powerful concept of the PU-integral, ibid. 38 (113) (1988), 8-48.
  5. [Ju] W. B. Jurkat, The Divergence Theorem and Perron integration with exceptional sets, ibid. 43 (118) (1993), 27-45.
  6. [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.
  7. [Kir] M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108.
  8. [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31 (106) (1981), 614-632.
  9. [McSh] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
  10. [No] D. J. F. Nonnenmacher, Theorie mehrdimensionaler Perron-Integrale mit Ausnahmemengen, PhD thesis, Univ. of Ulm, 1990.
  11. [Pf 1] W. F. Pfeffer, The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc. 43 (1987), 143-170.
  12. [Pf 2] W. F. Pfeffer, The Gauß-Green Theorem, Adv. in Math. 87 (1991), 93-147.
  13. [Pf 3] W. F. Pfeffer, A descriptive definition of a variational integral and applications, Indiana Univ. Math. J. 40 (1991), 259-270.
  14. [Pf-Ya] W. F. Pfeffer and W.-C. Yang, A multidimensional variational integral and its extensions, Real Anal. Exchange 15 (1989-1990), 111-169.
  15. [Rot] J. J. Rotman, An Introduction to Algebraic Topology, Graduate Texts in Math., Springer, 1988.
  16. [Saks] S. Saks, Theory of the Integral, Dover, New York, 1964.
  17. [Weir] A. J. Weir, General Integration and Measure, Vol. 2, Cambridge University Press, 1974.
  18. [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.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm145/fm14532.pdf

Fundamenta Mathematicae
1994/145/3
Export to PBN, Opens in new tab
Pages:
221-242
Main language of publication
English
Received
1992-09-13
Published
1994
Exact and natural sciences