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On Almost Uniform Convergence of Families of Functions

Published online by Cambridge University Press:  20 November 2018

Elias Zakon
Elias Zakon*
Affiliation:
University of Windsor
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In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 1 , March 1964 , pp. 45 - 48
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Egoroff, D. T., Sur les suites des fonctions mesurables. C.R. Acad. Sci. Paris, 152(1911).Google Scholar
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3. Halmos, P. R., Measure Theory. D. Van Nostrand, N. Y., 1950.Google Scholar
4. Munroe, M. E., Introduction to Measure and Integration. Addison-Wesley, Reading, Mass., 1959.Google Scholar
5. Tolstov, G., Une remarque sur le théorème de D. Th. Egoroff. Doklady Acad. Nauk SSSR (N. S.) 22(1939), 305–7.Google Scholar
6. Weston, J. D., A counterexample concerning Egoroff' s theorem. J. London Math. Soc. 34(1959), 139–140.Google Scholar