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Derivatives and Integrals with Respect to a Base Function of Generalized Bounded Variation

Published online by Cambridge University Press:  20 November 2018

H. W. Ellis  and
R. L. Jeffery
H. W. Ellis
Affiliation:
Summer Research Institute, Canadian Mathematical Congress
R. L. Jeffery
Affiliation:
Summer Research Institute, Canadian Mathematical Congress
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In this paper we consider measures determined by arbitrary functions G(x) for which finite right and left limits exist everywhere and indicate how some of these measures permit the definition of generalized integrals of constructive or Denjoy type. These definitions are related to corresponding descriptive definitions based on the Perron approach as given by Ward (6) and Henstock (2). An exposition of the introductory theory is given in (1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 225 - 241
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Ellis, H. W. and Jeffery, R. L., On measures determined by functions with finite right and left limits everywhere, submitted to Can. Math. Bull.Google Scholar
2. Henstock, R., A new descriptive definition of the Ward integral, J. London Math. Soc., 35 (1960), 43–48.Google Scholar
3. Henstock, R., N-variation, and N-variational integrals of set functions, Proc. London Math. Soc., 36 (1961), 109–132.Google Scholar
4. Jeffery, R. L., Theory of functions of a real variable, 2nd ed. (Toronto, 1953).Google Scholar
5. Munroe, M. E., Introduction to measure and integration (New York, 1953).Google Scholar
6. Ward, A. J., The Perron-Stieltjes integral, Math. Z., 41 (1936), 578–604.Google Scholar