Perron integral

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A generalization of the concept of the Lebesgue integral. A function is said to be integrable in the sense of Perron over if there exist functions (a major function) and (a minor function) such that

( and are the upper and lower derivatives) for , and if the lower bound to the values of the majorants is equal to the upper bound of the values of the minorants . Their common value is called the Perron integral of over and is denoted by

The Perron integral recovers a function from its pointwise finite derivative; it is equivalent to the narrow Denjoy integral. The Perron integral for bounded functions was introduced by O. Perron [1], while the final definition was given by H. Bauer [2].


[1] O. Perron, "Ueber den Integralbegriff" Sitzungsber. Heidelberg. Akad. Wiss. , VA (1914) pp. 1–16
[2] H. Bauer, "Der Perronsche Integralbegriff und seine Beziehung auf Lebesguesschen" Monatsh. Math. Phys. , 26 (1915) pp. 153–198
[3] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[4] I.A. Vinogradova, V.A. Skvortsov, "Generalized integrals and Fourier series" Itogi Nauk. Mat. Anal. 1970 (1971) pp. 67–107 (In Russian)


Perron's method is easier than Denjoy's, but Denjoy's method is more constructive. See (the editorial comments to) Denjoy integral.

For the definition of a major function and a minor function of see (the editorial comments to) Perron–Stieltjes integral.

How to Cite This Entry:
Perron integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article