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A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem

Published online by Cambridge University Press:  20 November 2018

W. B. Jurkat  and
R. W. Knizia
W. B. Jurkat
Affiliation:
Abteilung für Mathematik V, Universität Ulm, Oberer Eselsberg, 7900 Ulm (Donau), Federal Republic of Germany
R. W. Knizia
Affiliation:
Abteilung für Mathematik V, Universität Ulm, Oberer Eselsberg, 7900 Ulm (Donau), Federal Republic of Germany
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Abstract

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In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

Keywords

26A3926B3026A36Perron integralfundamental theoremfundamental inequalityabsolute continuityweak variationnull condition
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 3 , 01 June 1991 , pp. 526 - 539
Copyright
Copyright © Canadian Mathematical Society 1991

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