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Integration of Non-Measurable Functions

Published online by Cambridge University Press:  20 November 2018

Elias Zakon
Elias Zakon*
Affiliation:
University of Windsor, Canada
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This is primarily an expository paper based on (and generalizing) some ideas of J. Pierpont [6], W.H. Young [8], R. L. Jeffery [4] and S. C. Fan [2]. Our aim is to give a simple and easily applicable theory of integration for arbitrary extended - real functions over arbitrary sets in a measure space. This will be achieved by using a generalized version of Pierpont1 s upper and lower integrals (with the upper integral playing the main role), and by appropriately defining the operations in the extended real number system, henceforth denoted by E*, so as to make it a commutative semigroup under addition and multiplication.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 3 , August 1966 , pp. 307 - 330
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Dieudonné, J., Foundations of Modern Analysis. Academic Press, N.Y., 1960.Google Scholar
2. Fan, S. C., Integration with respect to an upper measure function, Amer. J. of Mathematics 63 (1941), 319-338.Google Scholar
3. Halmos, P.R., Measure Theory. D. Van No strand, Princeton 1959.Google Scholar
4. Jeffery, R. L., Relative summability, Ann. Math. 33 (1932), 443–459.Google Scholar
5. Munroe, M.E., Intro, to Measure and Integration. Addison- Wesley, 1959.Google Scholar
6. Pierpont, J., Theory of Functions of Real Var., vol.11, 1912.Google Scholar
7. Saks, S., Theory of the Integral, Monografie Matematyczne. Warsaw, 1937.Google Scholar
8. Young, W.H., On the general theory of integration, Philos. Trans. 204(1905), 221-252.Google Scholar