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On Ward's Perron-Stieltjes Integral

Published online by Cambridge University Press:  20 November 2018

Ralph Henstock
Ralph Henstock*
Affiliation:
Queen's University, Belfast
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In the paper (5), Ward defines an integral of Perron type of a finite function f with respect to another finite function g, where g need not be of bounded variation. There arise two problems, (a) and (b) below, that have not been dealt with in (5).

If f = j at a countable number of points everywhere dense in (a, b), where f and j are both integrable with respect to g, then f — j can be nonzero on a large set of points of (a, b).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 96 - 109
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Denjoy, A., Leçons sur le calcul des coefficients d'une série trigonométrique (Paris, 1941, 1949).Google Scholar
2. Henstock, R., The efficiency of convergence factors for functions of a continuous real variable, J. London Math. Soc, 30 (1955), 273–286.Google Scholar
3. Littlewood, J. E., The Elements of the Theory of Real Functions (Cambridge, 1926).Google Scholar
4. Saks, S., Theory of the Integral (Warsaw, 1937).Google Scholar
5. Ward, A. J., The Perron-Stieltjes integral, Math. Z., 41 (1936), 578–604.Google Scholar