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A Note on Measures Determined by Continuous Functions

Published online by Cambridge University Press:  20 November 2018

A. M. Bruckner
A. M. Bruckner*
Affiliation:
University of California, Santa Barbara, California
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Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 289 - 291
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Burry, J. H. W. and Ellis, H. W., On measures determined by continuous functions that are not of bounded variation, Canad. Math. Bull. (1) 13 (1970), 121-124.Google Scholar
2. Ellis, H. W. and Jeffery, R. L., On measures determined by functions with finite right and left limits everywhere, Canad. Math. Bull. (2) 10 (1967), 207-225.Google Scholar
3. Ellis, H. W. and Jeffery, R. L., Derivatives and integrals with respect to a base function of generalized variation, Canad. J. Math. 19 (1967), 225-241.Google Scholar
4. Marcus, S., Sur un théorème de M. S. Stoitow, concernant les functions continues d'une variable réelle, Rev. Gén. Sci. Pures Appl. 2 (1957), 409-412.Google Scholar
5. Munroe, M. E., Introduction to measure and integration, Addison-Wesley, Reading, Mass., 1953.Google Scholar
6. Saks, S., On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19(1932), 211-219.Google Scholar
7. Saks, S., Theory of the integral, Warsav, 1937 Google Scholar