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On Measures Determined by Continuous Functions that are not of Bounded Variation

Published online by Cambridge University Press:  20 November 2018

J. H. W. Burry  and
H. W. Ellis
J. H. W. Burry
Affiliation:
Queen's University, Kingston, Ontario
H. W. Ellis
Affiliation:
Queen's University, Kingston, Ontario
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In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 121 - 124
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. 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
2. Munroe, M. E., Introduction to measure and integration, Addison-Wesley, Cambridge, Mass. 1953.Google Scholar