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April, 1976 Some Finitely Additive Probability
Roger A. Purves, William D. Sudderth
Ann. Probab. 4(2): 259-276 (April, 1976). DOI: 10.1214/aop/1176996133


Lester E. Dubins and Leonard J. Savage have shown how to define a large family of finitely additive probability measures on the lattice of open sets of spaces of the form $X \times X \times \cdots$, where $X$, otherwise arbitrary, is assigned the discrete topology. This lattice does not include many of the sets which occur in the usual treatment of such probabilistic limit laws as the martingale convergence theorem, and in some unpublished notes Dubins and Savage conjectured that there might be a natural way to extend their measures to such sets. We confirm their conjecture here by showing that every set in the Borel sigma-field can be squeezed between an open and a closed set in the usual manner. It is then possible to generalize to this finitely additive setting many of the classical countably additive limit theorems. If assumptions of countable additivity are imposed, the extension studied here, when restricted to the usual product sigma-field, agrees with the conventional extension.


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Roger A. Purves. William D. Sudderth. "Some Finitely Additive Probability." Ann. Probab. 4 (2) 259 - 276, April, 1976.


Published: April, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0367.60034
MathSciNet: MR402888
Digital Object Identifier: 10.1214/aop/1176996133

Primary: 60G05
Secondary: 60A05 , 60B05 , 60F15 , 60G45

Keywords: dynamic programming , extension of measures , Finitely additive probability , finitely additive stochastic processes , Martingales , the strong law , theory of gambling , topological measures

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 2 • April, 1976
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