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)

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 $ f $ see (the editorial comments to) Perron–Stieltjes integral.