An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability

Timothy Myers

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 621-633
  • ISSN: 0011-4642

Abstract

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It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.

How to cite

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Myers, Timothy. "An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability." Czechoslovak Mathematical Journal 60.3 (2010): 621-633. <http://eudml.org/doc/38031>.

@article{Myers2010,
abstract = {It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.},
author = {Myers, Timothy},
journal = {Czechoslovak Mathematical Journal},
keywords = {absolute integrability; gauge Integral; H-K integral; Lebesgue integral; absolute integrability; gauge integral; H-K integral; Lebesgue integral},
language = {eng},
number = {3},
pages = {621-633},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability},
url = {http://eudml.org/doc/38031},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Myers, Timothy
TI - An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 621
EP - 633
AB - It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.
LA - eng
KW - absolute integrability; gauge Integral; H-K integral; Lebesgue integral; absolute integrability; gauge integral; H-K integral; Lebesgue integral
UR - http://eudml.org/doc/38031
ER -

References

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  1. Bartle, R., 10.2307/2974874, Am. Math. Mon. 103 (1996), 625-632 . (1996) Zbl0884.26007MR1413583DOI10.2307/2974874
  2. Bugajewska, D., On the equation of n th order and the Denjoy integral, Nonlinear Anal. 34 (1998), 1111-1115. (1998) Zbl0948.34003MR1637221
  3. Bugajewska, D., Bugajewski, D., On nonlinear integral equations and nonabsolute convergent integrals, Dyn. Syst. Appl. 14 (2005), 135-148 . (2005) Zbl1077.45004MR2128317
  4. Bugajewski, D., Szufla, S., On the Aronszajn property for differential equations and the Denjoy integral, Ann. Soc. Math. 35 (1995), 61-69 . (1995) Zbl0854.34005MR1384852
  5. Chew, T., Flordeliza, F., On x ' = f ( t , x ) and Henstock-Kurzweil integrals, Differ. Integral Equ. 4 (1991), 861-868 . (1991) Zbl0733.34004MR1108065
  6. Henstock, R., Definitions of Riemann type of the variational integral, Proc. Lond. Math. Soc. 11 (1961), 404-418. (1961) MR0132147
  7. Henstock, R., The General Theory of Integration, Oxford Math. Monogr., Clarendon Press, Oxford (1991) . Zbl0745.26006MR1134656
  8. Kurzweil, J., Generalized Ordinary Differential Equations and Continuous Dependence on a Parameter, Czech. Math. J. 7 (1957), 418-449 . (1957) Zbl0090.30002MR0111875
  9. McLeod, R., The Generalized Riemann Integral, Carus Math. Monogr., no. 20, Mathematical Association of America, Washington (1980) . Zbl0486.26005MR0588510
  10. Munkres, J., Analysis on Manifolds, Addison-Wesley Publishing Company, Redwood City, CA (1991) . Zbl0743.26006MR1079066
  11. Pfeffer, W., The divergence theorem, Trans. Am. Math. Soc. 295 (1986), 665-685 . (1986) Zbl0596.26007MR0833702
  12. Pfeffer, W., The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc. (Ser. A) 43 (1987), 143-170 . (1987) Zbl0638.26011MR0896622
  13. Rudin, W., Principles of Mathematical Analysis, Third Ed., McGraw-Hill, New York (1976) . Zbl0346.26002MR0385023
  14. Rudin, W., Real and Complex Analysis, McGraw-Hill, New York (1987). (1987) Zbl0925.00005MR0924157
  15. Schwabik, Š., The Perron integral in ordinary differential equations, Differ. Integral Equ. 6 (1993), 863-882 . (1993) Zbl0784.34006MR1222306
  16. Spivak, M., Calculus on Manifolds, W. A. Benjamin, Menlo Park, CA (1965). (1965) Zbl0141.05403MR0209411
  17. Stromberg, K., An Introduction to Classical Real Analysis, Waldworth, Inc (1981). (1981) Zbl0454.26001MR0604364

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