# 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 , while the final definition was given by H. Bauer .

How to Cite This Entry:
Perron integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron_integral&oldid=48165
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article