View source for Perron integral

<!--
p0723601.png
$#A+1 = 17 n = 0
$#C+1 = 17 : ~/encyclopedia/old_files/data/P072/P.0702360 Perron integral
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A generalization of the concept of the [[Lebesgue integral|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|Denjoy integral]]. The Perron integral for bounded functions was introduced by O. Perron [[#References|[1]]], while the final definition was given by H. Bauer [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,   "Ueber den Integralbegriff"  ''Sitzungsber. Heidelberg. Akad. Wiss.'' , '''VA'''  (1914)  pp. 1–16</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bauer,   "Der Perronsche Integralbegriff und seine Beziehung auf Lebesguesschen"  ''Monatsh. Math. Phys.'' , '''26'''  (1915)  pp. 153–198</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.A. Vinogradova,   V.A. Skvortsov,   "Generalized integrals and Fourier series"  ''Itogi Nauk. Mat. Anal. 1970''  (1971)  pp. 67–107  (In Russian)</TD></TR></table>

====Comments====
Perron's method is easier than Denjoy's, but Denjoy's method is more constructive. See (the editorial comments to) [[Denjoy integral|Denjoy integral]].

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