An axiomatic theory of non-absolutely convergent integrals in Rn

W. Jurkat; D. Nonnenmacher

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 3, page 221-242
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Jurkat, W., and Nonnenmacher, D.. "An axiomatic theory of non-absolutely convergent integrals in Rn." Fundamenta Mathematicae 145.3 (1994): 221-242. <http://eudml.org/doc/212044>.

@article{Jurkat1994,
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.},
author = {Jurkat, W., Nonnenmacher, D.},
journal = {Fundamenta Mathematicae},
keywords = {generalized Riemann integral; divergence theorem; non-absolutely convergent integrals; Saks-Henstock lemma},
language = {eng},
number = {3},
pages = {221-242},
title = {An axiomatic theory of non-absolutely convergent integrals in Rn},
url = {http://eudml.org/doc/212044},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Jurkat, W.
AU - Nonnenmacher, D.
TI - An axiomatic theory of non-absolutely convergent integrals in Rn
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 3
SP - 221
EP - 242
AB - 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.
LA - eng
KW - generalized Riemann integral; divergence theorem; non-absolutely convergent integrals; Saks-Henstock lemma
UR - http://eudml.org/doc/212044
ER -

References

top
  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 C 1 -transformations, Časopis Pěst. Mat. 109 (1984), 157-167. Zbl0555.26005
  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. Zbl0614.26007
  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. Zbl0669.26006
  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. Zbl0733.26008
  7. [Kir] M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108. Zbl60.0532.03
  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. Zbl0562.26004
  9. [McSh] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842. Zbl0010.34606
  10. [No] D. J. F. Nonnenmacher, Theorie mehrdimensionaler Perron-Integrale mit Ausnahmemengen, PhD thesis, Univ. of Ulm, 1990. Zbl0724.26010
  11. [Pf 1] W. F. Pfeffer, The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc. 43 (1987), 143-170. Zbl0638.26011
  12. [Pf 2] W. F. Pfeffer, The Gauß-Green Theorem, Adv. in Math. 87 (1991), 93-147. Zbl0732.26013
  13. [Pf 3] W. F. Pfeffer, A descriptive definition of a variational integral and applications, Indiana Univ. Math. J. 40 (1991), 259-270. Zbl0747.26010
  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. Zbl0286.28007
  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. Zbl0501.26006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.