The integral, a vision of its evolution through time II




Integral, Riemann- Stieltjes, Integral of Lebesgue, Integral of Young, Denjoy, Perron, Daniel, generalized R-integral, Henstock-Kurzweil


In this part II we intend to give an overview of the different ways in which the idea of integral has evolved over time; in particular the Riemann integral has been the motivation for further research up to the present.

We only want to motivate this part of the history of mathematical analysis in our environment with the aim that student and teachers have a guide for more detailed studies and thus contribute to the development of matematics in our region.


Bartle RG. Return to the Riemann Integral. American Mathematical Monthly. 103.8. 1980.

Bongiorno B. On the C-integral, Dpto Mathematics University of Palerme. Arch. 34, 90123. 2002.

Bourbaki N. Elementos de historia de las matemáticas. Alianza Editorial. Madrid. 1972.

Brito W. Las integrales de Riemann, Lebesgue y Hernstock- Kurzweil[Internet], ULA-Venezuela; [accesado en 19 febrero 2023].Disponible en

Burkill JC. The Lebesgue Integral. Cambridge University Press. London. 1953.

Cross G, Oved S. A new approach to integration. Journal of Mathematical Analysis and Applications. 114. 1986.

Chatterji SD. Remarques sur l'integrale de Riemann g´en´eralis´ee. Seminaire de Probabilites de Strasbourg. Vol. 30.1996.

Daniell, Percy: A general form of integral. Ann. Math. 1917-18.

Darboux, G: Mémoire Sur les fontions discontinues. Ann. Ecole. Norm. Sup. 1875; 4(2).

Denjoy A. Une extension de l'integrale de M.Lebesgue. Compt. Rend. 1912; 154.

Henstock R. Definitions of Riemann type of the variational integrals. Proc. London Math. Soc. 1961; (3)11.

Herrera JE. La integral de Henstock-Kurzweil[Internet]. Universidad de Panmá. 2005. Disponible en

Katznelson Y. An introduction to harmonic analysis. Dover Public. New York. 1976.

Khinchin A. Sur une extension de l’integrale de M.Denjoy. Compt. Rend.162. 1916

Khinchin A. Sur le proc´ed´e d’integration de M.Denjoy. Rec. Math.Soc.Math. Moscow, 1918; 30.

Kurtz D, Swartz Ch. Theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil and Mcshane. World Scientific Publ. 2004.

Kurzweil J. Generelized Ordinary Differential equations and continous dependence on a parameter. Czechos. Math. J. 1957; Vol.7. No 3.

Lewis J, Shisha O. The generalized Riemann, domined improper integrals. Journal of Approx. Theory. 1983; 38.

Medvedev FA. Scenes from the history of real functions. Birkh¨auser Verlag. Boston. 1991.

Muldowney P. (Editor): Henstock lectures on integration theory. New University of Ulsler. 1970-1971.

Nachbin L. Integral de Haar. Textos de Matemáticas. Instituto de Física e Matemática, Univ. Recife. 1960.

Ortiz A. Aspectos básicos en ecuaciones en derivadas parciales. Notas de Matemática No 3. Dpto. Matem. UNT. 1988.

Ortiz A. La integral, una visión de su evolución a través del tiempo I. Selecciones Matemáticas. Vol 10(1).173-198. Universidad Nacional de Trujillo. 2023.

Pesin IN. Classical and Modern Integration Theories, Academic Press. N.Y. 1970.

Riesz F, Nagy B. Functional analysis. Ungar, New York.1955.

Saab E. Unified integration”by E.J. Mc Shane. 1983. Bulletin of the AMS. 1985; Vol. 13. No 1.

Saks S. Theory of the integral. 2da Edic. Dover Publications, INC. New York. 1964.

Shisha O. The genesis of the generalized Riemann integral. Computers Math. Applic. 1995; Vol. 30. No 3-6.

Thomson B. On VBG functions and the Denjoy- Khinchin integral. Real Analysis. Exchange. 2015/2016; Vol.41(1).

Torchinsky A. Real Variables. U. Indiana. Addison-Wesley. Publis. Comp, 1988.

Wheeden RL, Zygmund A. Measure and integral. An introduction to real analysis. Marcel Dekker. INC.New Yrok. 1977.

Young WH. On the general theory of integration. Phil. Trans. Roy. Soc.London 204A. 1905.

Young WH. On the new general theory of integration. Proc. Roy. London 88A. 1912.

Young WH. Integration with respect to a function of bounded variation. Proc. London. Math. Soc. 1914; 13(2).

Natanson IP. Theory of functions of a real variable, Vols. I-II. Frederick Ungar Publishing. 1955.



How to Cite

Ortiz Fernández , A. (2023). The integral, a vision of its evolution through time II. Selecciones Matemáticas, 10(02), 381 - 403.