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A Remark on the Theorems of Lusin and Egoroff

Published online by Cambridge University Press:  20 November 2018

Elias Zakon
Elias Zakon*
Affiliation:
University of Windsor
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In this note we do not intend to establish new results but only to suggest a very simple proof of Lusin's theorem, direct for σ-finite regular measures, a proof that bypasses the usual procedure of first establishing this theorem for sets of finite measure only. The proposed proof utilizes the notion of subuniform convergence, a method which seems not yet to have been used, despite its simplicity and adaptability. Simultaneously, a useful supplement to Egoroff's theorem will be obtained.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 2 , June 1964 , pp. 291 - 295
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Egoroff, D. T., Sur les suites des fonctions mesurables, C.R. Acad. Sci. Paris, 152(1911), 244–246.Google Scholar
2. Halmos, P. R., Measure Theory, D. Van Nostrand, N.Y., 1950.Google Scholar
3. Lusin, N., Sur les propriétés des fonctions mesurables, C.R. Acad. Sci. Paris, 154(1912), 1688–1690.Google Scholar
4. Munroe, M. E., Introduction to Measure and Integration, Addison-Wesley, Reading, Mass., 1959.Google Scholar
5. Saks, S., Theory of the Integral, Hafner, N.Y., 1937.Google Scholar
6. Schaerf, H. M., On the continuity of measurable functions in neighborhood spaces, Portug. Mathem., (1947), 33–44.Google Scholar
7. Schaerf, H. M., Dtto (II), ibid., (1948), 91–92.Google Scholar
8. Kelley, J., General Topology, Van Nostrand, 1960.Google Scholar