Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T16:12:19.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Perron Integrability Versus Lebesgue Integrability

Published online by Cambridge University Press:  20 November 2018

Arlo W. Schurle
Arlo W. Schurle*
Affiliation:
Department of Mathematical Sciences, University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper investigates the relationship between Perron - Stieltjes integrability and Lebesgue-Stieltjes integrability within the generalized Riemann approach. The main result states that with certain restrictions a Perron-Stieltjes integrable function is locally Lebesgue-Stieltjes integrable on an open dense set. This is then applied to show that a nonnegative Perron-Stieltjes integrable function is Lebesgue-Stieltjes integrable. Finally, measure theory is invoked to remove the restrictions in the main result.

Keywords

26A3926A42
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 463 - 468
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Davis, R.O. and Schuss, Z., A proof that Henstock's integral includes Lebesgue's, J. London Math. Soc. (2)2, (1970), pp. 561–562.Google Scholar
2. Henstock, R., The efficiency of convergence factors for functions of a continuous real variable, J. London Math. Soc. 30 (1955), pp. 273–286.Google Scholar
3. Henstock, R., Theory of Integration, Butterworths, London, 1963.Google Scholar
4. Henstock, R., A Riemann-type integral of Lebesgue power, Canad. J. Math. 20, (1968), pp. 79–87.Google Scholar
5. Kubota, Y., A direct proof that the RC-integral is equivalent to the D*-integral, Proc. Amer. Math. Soc. 80, (1980), pp. 293–296.Google Scholar
6. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7, (1957), pp. 418–446.Google Scholar
7. McLeod, R.M., The Generalized Riemann Integral, Cams Mathematical Monograph No. 20, 1980, American Mathematical Association, Washington, D.C. Google Scholar
8. McShane, E.J., A Riemann-type integral that includes Lebesgue—Stieltjes, Bochner and stochastic integrals, Memoirs Amer. Math. Soc, No. 88, 1969.Google Scholar
9. Natanson, I.P., Theory of Functions of a Real Variable, Vol. II, translated by Boron, L.F., Frederick Ungar Publishing Co., New York, 1960.Google Scholar
10. Pfeffer, W.F., The Riemann-Stieltje s Approach to Integration, Technical Report 187, National Research Institute for Mathematical Sciences, Pretoria, South Africa.Google Scholar
11. Saks, S., Theory of the Integral, (2nd revised edition), translated by Young, L.C., Dover Publications, New York, 1964.Google Scholar