Henstock integrable functions are Lebesgue integrable on a portion. (English) Zbl 0732.26011

Buczolich, Zoltán

Proc. Am. Math. Soc. 111, No. 1, 127-129 (1991).

This paper provides an affirmative answer to a question raised in 1955 by Karták: for a Perron integrable function, can one find a nondegenerate interval on which it is Lebesgue integrable? The author uses the Kurzweil-Henstock definition of the Perron integral in n dimensions, Riemann sums and Baire category theorem to prove that if \(I\subset {\mathbb{R}}^ n\) is an interval and f is Kurzweil-Henstock integrable on I, then there exists a nondegenerate interval \(J\subset I\) such that f is Lebesgue integrable on J.

Reviewer: J.Mawhin (Louvain-La-Neuve)

MSC:

26B15	Integration of real functions of several variables: length, area, volume
26A39	Denjoy and Perron integrals, other special integrals
26A42	Integrals of Riemann, Stieltjes and Lebesgue type

Keywords:

Kurzweil-Henstock integral; Lebesgue integral; Perron integrable function; Perron integral

References:

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[5]	S. Saks, Theory of the integral, Hafner, New York, 1937. · Zbl 0017.30004