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On the Lower Derivate of a Set Function

Published online by Cambridge University Press:  20 November 2018

W. F. Pfeffer
W. F. Pfeffer*
Affiliation:
University of California, Davis, California
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In (5), the following theorem was proved in a very general setting:

(1) An additive set function is non-negative whenever its lower derivative is non-negative.

For a continuous additive function of intervals, theorem (1) can be improved as follows:

(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1489 - 1498
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Bogdanowicz, W. M., A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 492–498.Google Scholar
2. Bruckner, A. M. and Leonard, J. L., Derivatives, Amer. Math. Monthly 73 (1966), no. 4, part II, 24–56.Google Scholar
3. Kamke, E., Das Lebesgue-Stieltjes-Integral (Teubner, Verlagsgesellschaft, Leipzig, 1956).Google Scholar
4. Kelley, J. L., General topology (Van Nostrand, New York, 1955).Google Scholar
5. Pfeffer, W. F., A note on the lower derivative of a set function and semihereditary systems of sets, Proc. Amer. Math. Soc. 18 (1967), 1020–1025.Google Scholar
6. Pfeffer, W. F., The integral in topological spaces (to appear in J. Math. Mech.).Google Scholar
7. Smirnov, Yu. M., A necessary and sufficient condition for metrizability of a topological space, Dokl. Akad. Nauk SSSR 77 (1951), 197–200.Google Scholar