Perron integral

A generalization of the concept of the Lebesgue integral. A function $f$ is said to be integrable in the sense of Perron over $[ a, b]$ if there exist functions $M$( a major function) and $m$( a minor function) such that

$$M( a) = 0,\ \ \underline{D} M ( x) \geq f( x),\ \ \underline{D} M( x) \neq - \infty ,$$

$$m( a) = 0,\ \overline{D}\; m( x) \leq f( x),\ \overline{D}\; m( x) \neq + \infty$$

( $\underline{D}$ and $\overline{D}\;$ are the upper and lower derivatives) for $x \in [ a, b]$, and if the lower bound to the values $M( b)$ of the majorants $M$ is equal to the upper bound of the values $m( b)$ of the minorants $m$. Their common value is called the Perron integral of $f$ over $[ a, b]$ and is denoted by

$$( P) \int\limits _ { a } ^ { b } f( x) dx.$$

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)

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