# 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)