Additive set functions on lattices of sets

DeBoth, Gene A.

doi:10.1090/S0002-9947-1973-0333109-X

Additive set functions on lattices of sets
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by Gene A. DeBoth PDF

Trans. Amer. Math. Soc. 178 (1973), 341-355 Request permission

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.

References

H. D. Brunk, Conditional expectation given a $\sigma$-lattice and applications, Ann. Math. Statist. 36 (1965), 1339–1350. MR 185629, DOI 10.1214/aoms/1177699895
H. D. Brunk and S. Johansen, A generalized Radon-Nikodym derivative, Pacific J. Math. 34 (1970), 585–617. MR 267066, DOI 10.2140/pjm.1970.34.585
Richard B. Darst, The Lebesgue decomposition, Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets, Pacific J. Math. 35 (1970), 581–600. MR 291398, DOI 10.2140/pjm.1970.35.581

Norm convergence of martingales of Radón-Nikodym derivatives given a

lattice

Two approximation properties and a Radón Nikodym derivative for lattices of sets

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
S. Johansen, The descriptive approach to the derivative of a set function with respect to a $\sigma$-lattice, Pacific J. Math. 21 (1967), 49–58. MR 209427, DOI 10.2140/pjm.1967.21.49
M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
Solomon Leader, The theory of $L^p$-spaces for finitely additive set functions, Ann. of Math. (2) 58 (1953), 528–543. MR 58126, DOI 10.2307/1969752
B. J. Pettis, On the extension of measures, Ann. of Math. (2) 54 (1951), 186–197. MR 41912, DOI 10.2307/1969320
J. J. Uhl Jr., Applications of Radon-Nikodým theorems to martingale convergence, Trans. Amer. Math. Soc. 145 (1969), 271–285. MR 251756, DOI 10.1090/S0002-9947-1969-0251756-X
J. J. Uhl Jr., Martingales of vector valued set functions, Pacific J. Math. 30 (1969), 533–548. MR 248515, DOI 10.2140/pjm.1969.30.533
J. J. Uhl Jr., Orlicz spaces of finitely additive set functions, Studia Math. 29 (1967), 19–58. MR 226395, DOI 10.4064/sm-29-1-19-58