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A generalization of the concept of the [[Lebesgue integral|Lebesgue integral]]. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723601.png" /> is said to be integrable in the sense of Perron over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723602.png" /> if there exist functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723603.png" /> (a major function) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723604.png" /> (a minor function) such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723605.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723606.png" /></td> </tr></table>
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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
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723608.png" /> are the upper and lower derivatives) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p0723609.png" />, and if the lower bound to the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236010.png" /> of the majorants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236011.png" /> is equal to the upper bound of the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236012.png" /> of the minorants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236013.png" />. Their common value is called the Perron integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236015.png" /> and is denoted by
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$$
 +
M( a)  = 0,\ \
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\underline{D} M ( x)  \geq  f( x),\ \
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\underline{D} M( x) \neq  - \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236016.png" /></td> </tr></table>
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$$
 +
m( a)  = 0,\  \overline{D}\; m( x)  \leq  f( x),\  \overline{D}\; m( x)  \neq  + \infty
 +
$$
 +
 
 +
( $  \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]]].
 
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]]].
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====References====
 
====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>
 
<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====
 
====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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072360/p07236017.png" /> see (the editorial comments to) [[Perron–Stieltjes integral|Perron–Stieltjes 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]].

Latest revision as of 08:05, 6 June 2020


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=11984
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article