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The representation of linear operators on spaces of finitely additive set functions

Published online by Cambridge University Press:  20 January 2009

C. A. Cheney  and
Andre de Korvin
C. A. Cheney
Affiliation:
Department of Mathematics, Indiana State University, Terre Haute, Indiana 47809
Andre de Korvin
Affiliation:
Department of Mathematics, Indiana State University, Terre Haute, Indiana 47809
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The theory of representing continuous linear operators on function spaces in terms of integrals has had a long and fruitful history, beginning with the Riesz representation theorem in 1909. If T is such an operator, then the standard representation is T (f) = ∫f dμ, where the integral is denned in diverse ways, depending on the nature of the set of functions and the nature of T.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 3 , March 1977 , pp. 233 - 242
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

REFERENCES

(1) Alo, R. A. and De Korvin, A., Representation theorems for linear operators, Proceedings of the Conference on Functional Analysis and its Applications,Kanpur, India, pp. 1–12.Google Scholar
(2) Alo, R. A., De Korvin, A. and Easton, R., Functions of bounded variation on idempotent semigroups, Math. Ann. 194 (1971), 1–11.CrossRefGoogle Scholar
(3) Darst, R., A decomposition for finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31–37.CrossRefGoogle Scholar
(4) De Korvin, A. and Easton, R., Some representation theorems, Rocky Mountain J. Math. 1 (1971), 561–572.Google Scholar
(5) Dunford, N. and Schwartz, J., Linear Operators, Part I (New York, Interscience, 1958).Google Scholar
(6) Edwards, J. and Wayment, S., Extensions of the v-integral, Trans. Amer. Math. Soc. 191 (1974), 165–184.Google Scholar
(7) Fefferman, C., A Radon-Nikodym theorem for finitely additive set functions, Pacific J. Math. 23 (1967), 35–45.CrossRefGoogle Scholar
(8) Gordon, H., The maximal ideal space of a ring of measurable functions, Amer. J. Math. 88 (1966), 827–843.CrossRefGoogle Scholar
(9) Mauldin, D., A representation for the second dual of C[0,1], Studia Math. (to appear).Google Scholar
(10) Singer, I., Bases in Banach Spaces (Springer-Verlag, New York, 1970).CrossRefGoogle Scholar
(11) Uhl, J., Extensions and decompositions of vector measures, J. London Math. Soc. 3 (1971), 672–676.CrossRefGoogle Scholar