Rendiconti del Circolo Matematico di Palermo

, Volume 21, Issue 3, pp 293–304 | Cite as

On the existence of ψ-integrals

  • James D. Baker
  • Robert A. Shive
Article

Summary

The approximating sums for the Riemann-Stieltjes integral are formed with the integrand function evaluated at an arbitrary point in each subdivision interval. The ψ-integral uses the idea of achoice function to select a particular point in each of these intervals.

In this paper, we investigate existence conditions for a Stieltjes type integral on the line. Rather general necessary conditions aregiven for the existence of this integral, and necessary and sufficient conditions are given for the case when one of the integrand or integrator functions is quasi-continuous and the other is of bounded variation. Then by fixing one of the integrand or integrator functions and by selecting an appropriate choice function, we determine how large the other class of functions can be and have the integral still exist.

Keywords

Integrator Function Step Function Bounded Variation Choice Function Linear Topological Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Kristensen, E., Poulsen, E. T., and Reich, E.,A characterization of Riemann-integrability, Amer. Math. Monthly, 69 (1962), 498–505.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Fieger, W.,Riemannintegrabilitat and ψ-integrabilitat, Math. Ann, 164 (1966), 30–37.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Marcus, S.,Mathematical Reviews, 25 (1963), 4074.Google Scholar
  4. [4]
    Dyer, J. A.,Integral bases in linear topological spaces, Illinois J. Math., 14 (1970), 468–477.MATHMathSciNetGoogle Scholar
  5. [5]
    Woodworth, W.,Integrals over proto-rings, Ph. D. dissertation. Ames, Iowa, Library, Iowa State University, 1968.Google Scholar
  6. [6]
    Shive, R. A., Jr.,Existence theorems for the ψ-integral, Ph. D. dissertion. Ames, Iowa, Library, Iowa State University, 1969.Google Scholar
  7. [7]
    Hildebrandt, T. H.,Definitions of Stieltjes integrals of the Riemann type, Amer. Math. Monthly, 45 (1938), 265–278.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Lane, R. E.,The integral of a function with respect to a function, Amer. Math. Soc. Proc., 5 (1954), 59–66.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Eltze E. M.,Some properties of ψ-integration, Ph. D. dissertation. Ames, Iowa, Library, Iowa State University, 1970.Google Scholar

Copyright information

© Springer 1972

Authors and Affiliations

  • James D. Baker
    • 1
  • Robert A. Shive
    • 2
  1. 1.BloomingtonU.S.A.
  2. 2.JacksonU.S.A.

Personalised recommendations