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A Lebesgue Decomposition for Vector Valued Additive Set Functions Defined on a Lattice

Published online by Cambridge University Press:  20 November 2018

Thomas P. Dence
Thomas P. Dence*
Affiliation:
Bowling Green State University, Huron, Ohio
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Abstract

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Our aim is to establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets in both the finitely and countably additive setting. Strongly bounded (s-bounded) set functions were first studied by Rickart [15], and then by Rao [14], Brooks [1] and Darst [5; 6]. In 1963 Darst [6] established a result giving the decomposition of s-bounded elements in a normed Abelian group with respect to an algebra of projection operators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 2 , 01 April 1977 , pp. 295 - 298
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Brooks, James, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001.Google Scholar
2. Brunk, H. D., Best fit to a random variable by a random variable measurable with respect to a a-lattice, Pac. J. Math. 11 (1961), 785–802.Google Scholar
3. Brunk, H. D. Conditional expectation given a a-lattice and applications, Ann. Math. Statist. 36 (1965), 1339–1350.Google Scholar
4. Brunk, H. D. and Johansen, S., A generalized Radon-Nikodym derivative, Pac. J. Math. 84 (1970), 585–617.Google Scholar
5. Darst, R. B., A decomposition for complete normed Abelian groups with applications to spaces of additive set functions, Trans. Amer. Soc, 103 (1962), 545–559.Google Scholar
6. Darst, R. B. The Lebesgue decomposition, Duke Math. J. 30 (1963), 553–556.Google Scholar
7. Darst, R. B. The Lebesgue decomposition for lattices of projection operators, Advances in Math. 17 (1975), 30–33.Google Scholar
8. Darst, R. B. and DeBoth, G. A., Two approximation properties and a Radon-Nikodym derivative for lattices of sets, Indiana U. Math. J. 21 (1971), 47–58.Google Scholar
9. Darst, R. B. and DeBoth, G. A. Norm convergence of martingales of Radon-Nikodym derivatives given a a-lattice, Pac. J. Math. 40 (1972), 547–552.Google Scholar
10. Drewnowski, L., Decompositions of set functions, Studia Math. 48 (1973), 218–231.Google Scholar
11. Halmos, Paul, Measure theory (Van Nostrand, 1950).Google Scholar
12. Johansen, S., The descriptive approach to the derivative of a set function with respect to a a-lattice, Pac. J. Math. 21 (1967), 49–58.Google Scholar
13. Pettis, B. J., On the extension of measures, Annals of Math. 54 (1951). 186–197.Google Scholar
14. Rao, M. M., Decomposition of vector measures, Proc. Nat. Acad. Sci. USA 51 (1964), 771–774.Google Scholar
15. Rickart, C. E., Decomposition of additive set functions, Duke Math. J. 10 (1943), 653–665.Google Scholar