# 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.

**How to Cite This Entry:**

Perron integral.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Perron_integral&oldid=48165