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Necessary and sufficient conditions for the existence of a generalized Stieltjes integral

Published online by Cambridge University Press:  09 April 2009

A. M. Russell
A. M. Russell
Affiliation:
Department of Mathematics University of Melbourne Parkville 3052 Australia
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Abstract

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In Russell (1973) a Riemann-type necessary and sufficient condition was given for the existence of (defined also in Russell (1975)) when f was bounded and g was k-convex in [a′, b′]. In this paper we present necessary and sufficient conditions for the existence of a particular Stieltjes-type integral without imposing a convexity condition upon g. These conditions are used to obtain an additivity result for the integral over adjoining intervals without any additional restrictions being imposed upon the functions involved.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 4 , December 1978 , pp. 501 - 510
Copyright
Copyright © Australian Mathematical Society 1978

References

Hildebrandt, T. H. (1963), Introduction to the theory of integration (pure and Appl. Math Vol. 13, Academic Press, New York).CrossRefGoogle Scholar
Russell, A. M. (1973), ‘Functions of bounded kth variation, and Stieltjes-type integrals’ (Doctoral dissertation, University of Melbourne).Google Scholar
Russell, A. M. (1975), ‘Stieltjes-type integrals’, J. Austral. Math. Soc. (Ser A) 20, 431–448.CrossRefGoogle Scholar