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Majorants in Variational Integration

  • Ralph Henstock (a1)

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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.

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References

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1. Henstock, R., A new descriptive definition of the Ward integral, J. London Math. Soc, 35 (1960), 4348.
2. Henstock, R., N-variation and N-variational integrals of set functions, Proc. London Math. Soc. (3), 11 (1961), 109133.
3. Henstock, R., The theory of integration (London, 1963).
4. Karták, K., K theorii vícerozměrného integrálu, Časopis Pest. Mat., 80 (1955), 400414 (Russian and German summary).
5. Mařik, J., Základy theorie integrálu v euklidovyćh prostorech, Časopis Pest. Mat., 77 (1952), 151, 125-145, 267-301.
6. Saks, S., Theory of the integral (2nd English edition, Warsaw, 1937).
7. Ward, A. J., The Perron-Stieltjesintegral, Math. Z., 41 (1936), 578604.
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