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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Additive set functions on lattices of sets


Author: Gene A. DeBoth
Journal: Trans. Amer. Math. Soc. 178 (1973), 341-355
MSC: Primary 28A15; Secondary 60G45
MathSciNet review: 0333109
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Abstract: This paper is concerned with properties of additive set functions defined on lattices of sets. Extensions of results of Brunk and Johansen, Darst, Johansen, and Uhl are obtained. Two fundamental approximation properties for lattices of sets (established in another paper) permit us to translate the setting and consider countably additive set functions defined on sigma lattices of sets. Thereby results for countably additive set functions defined on sigma lattices of sets are used to obtain alternate derivations and extensions of Darst's results for additive set functions defined on lattices of sets, i.e., we consider the Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0333109-X
PII: S 0002-9947(1973)0333109-X
Keywords: Lattice of sets, algebra of sets, finitely additive set functions, sigma algebra, sigma lattice, $ {\Delta _2}$-condition, Orlicz space, Radon-Nikodym derivative, conditional expectation, martingale
Article copyright: © Copyright 1973 American Mathematical Society