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Majorants in Variational Integration

Published online by Cambridge University Press:  20 November 2018

Ralph Henstock
Ralph Henstock*
Affiliation:
The University of Lancaster, Lancaster, England
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In Perron integration, majorants are usually functions of points. If the domain of definition is a Euclidean space of n dimensions, we can define a finitely additive n-dimensional majorant rectangle function by taking suitable differences of the majorant point function with respect to each of the n coordinates. The way is then open to a generalization, in that we need only suppose that the majorant rectangle function is finitely superadditive. Similarly, we need only suppose that a minorant rectangle function is finitely subadditive. These kinds of rectangle functions were used by J. Mařík (5) to prove the Fubini theorem for Perron integrals in Euclidean space of m + n dimensions. He also proved that for a function that is Perron, and absolutely Perron, integrable, the majorant and minorant rectangle functions can be taken to be finitely additive. As a result he posed the following problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 49 - 74
Copyright
Copyright © Canadian Mathematical Society 1966

References

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