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Difference between revisions of "Khinchin integral"

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A generalization of the narrow [[Denjoy integral|Denjoy integral]] introduced by A.Ya. Khinchin in [[#References|[1]]].
 
A generalization of the narrow [[Denjoy integral|Denjoy integral]] introduced by A.Ya. Khinchin in [[#References|[1]]].
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055380/k0553801.png" /> is said to be integrable in the sense of Khinchin on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055380/k0553802.png" /> if it is Denjoy-integrable in the wide sense and if its indefinite integral is differentiable almost everywhere. Sometimes the Khinchin integral is also called the Denjoy–Khinchin integral.
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A function $f$ is said to be integrable in the sense of Khinchin on $[a,b]$ if it is Denjoy-integrable in the wide sense and if its indefinite integral is differentiable almost everywhere. Sometimes the Khinchin integral is also called the Denjoy–Khinchin integral.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "Sur une extension de l'intégrale de M. Denjoy"  ''C.R. Acad. Sci. Paris'' , '''162'''  (1916)  pp. 287–291</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya. Khinchin,  ''Mat. Sb.'' , '''30'''  (1918)  pp. 543–557</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.N. Pesin,  "Classical and modern integration theories" , Acad. Press  (1970)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "Sur une extension de l'intégrale de M. Denjoy"  ''C.R. Acad. Sci. Paris'' , '''162'''  (1916)  pp. 287–291</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya. Khinchin,  ''Mat. Sb.'' , '''30'''  (1918)  pp. 543–557</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.N. Pesin,  "Classical and modern integration theories" , Acad. Press  (1970)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>

Latest revision as of 08:39, 23 July 2014

A generalization of the narrow Denjoy integral introduced by A.Ya. Khinchin in [1].

A function $f$ is said to be integrable in the sense of Khinchin on $[a,b]$ if it is Denjoy-integrable in the wide sense and if its indefinite integral is differentiable almost everywhere. Sometimes the Khinchin integral is also called the Denjoy–Khinchin integral.

References

[1] A.Ya. Khinchin, "Sur une extension de l'intégrale de M. Denjoy" C.R. Acad. Sci. Paris , 162 (1916) pp. 287–291
[2] A.Ya. Khinchin, Mat. Sb. , 30 (1918) pp. 543–557
[3] I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian)
[4] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
How to Cite This Entry:
Khinchin integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Khinchin_integral&oldid=32527
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article