Indian Journal of Pure and Applied Mathematics

, Volume 42, Issue 6, pp 493–509 | Cite as

Further randomization of Riemann sums leading to the Lebesgue integral

  • A. Varsei
  • E. Dastranj


In this paper we randomize in a particular way the sequence of partitions based on which the random Riemann sums are defined for a Lebesgue integrable function f on (0, 1). Convergence of such sums to the Lebesgue integral of f is investigated.

Key words

Random partitions Riemann sums random Riemann sums Lebesgue integral 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Abramowitz and I. A. Stegun (Eds)., Stirling Numbers of the Second Kind, 24.1.4 in Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 5th printing. New York: Dover, (1972), 824–825.Google Scholar
  2. 2.
    J. Grahl, A Random Approach to the Lebesgue Integral, J. Math. Anal. Appl., 340 (2008), 358–365.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Charles S. Kahane, Evaluating Lebesgue Integrals as Limits of Riemann Sums, Math. Japonica, 38 (1993), 1073–1076.MathSciNetzbMATHGoogle Scholar
  4. 4.
    O. Kallenberg, Random Measures, Academic Press, (1983).Google Scholar
  5. 5.
    John C. Kieffer and Caslav V. Stanojevic, the Lebesgue Integral as the Almost Sure Limit of Random Riemann Sums, Proc. Amer. Math. Soc., 85(3) (1982), 389–392.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Alexander R. Pruss, Randomly Sampled Riemann Sums and Complete Convergence in the Law of Large Numbers for a CaseWithout Identical Distribution, Proc. Amer. Math. Soc., 124 (1996), 919–929.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Indian National Science Academy 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of GuilanRashtIran

Personalised recommendations