# Perron integral

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].

#### References

[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) |

#### Comments

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: http://encyclopediaofmath.org/index.php?title=Perron_integral&oldid=48165