Harnack integral

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A generalization of the improper Riemann integral on the class of functions $ f $ whose set of unboundedness points $ E _ {f} $ has Jordan measure zero, and which are Riemann integrable on any segment not containing points of $ E _ {f} $. Let $ \Delta _ {i} $, $ i = 1 \dots n $, be a finite system of intervals containing $ E _ {f} $. The Harnack integral is then defined by the equation

$$ ( H) \int\limits _ { a } ^ { b } f ( x) dx = \ \lim\limits ( R) \int\limits _ {( a, b) \setminus \cup _ {i} \Delta _ {i} } f ( x) dx, $$

if the last limit for $ \mathop{\rm mes} ( \cup _ {i} \Delta _ {i} ) \rightarrow 0 $ exists. The integral was introduced by A. Harnack [1]. The condition that each interval $ \Delta _ {i} $ should have non-empty intersection with $ E _ {f} $ was added to the original definition at a later date. As a result, the Harnack integral usually becomes conditionally convergent. It partly overlaps the Lebesgue integral and is covered by the Perron integral and by the Denjoy integral. Nowadays, the Harnack integral is only of methodical and historical interest.


[1] A. Harnack, "Anwendung der Fourier'schen Reihe auf die Theorie der Funktionen einer komplexen Veränderlichen" Math. Ann. , 21 (1883) pp. 305–326
[2] I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian)



[a1] E.W. Hobson, "The theory of functions of a real variable and the theory of Fourier's series" , 1 , Dover, reprint (1957)
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Harnack integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article