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The Vitali Integral Convergence Theorem and Uniform Absolute Continuity

Published online by Cambridge University Press:  20 November 2018

Elizabeth M. Bator ,
Russell G. Bilyeu  and
Paul W. Lewis
Elizabeth M. Bator
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas, USA
Russell G. Bilyeu
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas, USA
Paul W. Lewis
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas, USA
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Abstract

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A geometric version of the Vitali integral convergence theorem is established. Parameterized versions of results on uniform absolute continuity in spaces of measures suggested by the convergence theorem are studied.

Keywords

46B2046A5028A3328A99Vitali integral convergence theoremuniform absolute continuityGateaux differentiability
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 5 , 01 October 1991 , pp. 939 - 947
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bartle, R.G., Dunford, N., and Schwartz, J.T., Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305.Google Scholar
2. Bilyeu, R.G. and Lewis, P.W., Uniform differentiability, uniform absolute continuity, and the Vitali-Hahn- Saks theorem, Rocky Mtn. J. Math. 10 (1980), 533–557.Google Scholar
3. Brooks, J.K., Equicontinuous sets of measures and applications to Vitali's integral convergence theorem and control measures, Advances in Math. 10 (1973), 165–171.Google Scholar
4. Brooks, J.K. and Jewett, R.S., On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298.Google Scholar
5. Diestel, J., Geometry of Banach spaces—selected topics. Lecture Notes in Math., 485, Springer-Verlag, New York, 1975.Google Scholar
6. Rockafellar, R.T., Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16(1969), 397–407.Google Scholar
7. Rockafellar, R.T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 200–216.Google Scholar
8. Royden, H.L., Real analysis. The Macmillian Company, Toronto, Canada.Google Scholar