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June, 1969 Conditional Probability on $\Sigma$-Complete Boolean Algebras
Ardel J. Boes
Ann. Math. Statist. 40(3): 970-978 (June, 1969). DOI: 10.1214/aoms/1177697601


Probability as measure on a Boolean algebra was presented by Kappos [5], but a treatment of conditional probability relative to a subalgebra is missing. The Stone space of a $\sigma$-complete Boolean algebra (see [10], p. 24) enables one to apply the concepts of conditional probability for a $\sigma$-algebra of subsets of some space (see [2], pp. 15-28), but the problem deserves closer attention. In this note we consider conditional probability with respect to a $\sigma$-subfield of the $\sigma$-field generated by the open-closed subsets of the Stone space of a Boolean $\sigma$-algebra. We show that there is always a regular conditional probability (see [4], p. 80) relative to a full $\sigma$-subalgebra of Baire sets. With a modified definition of probability on a Boolean algebra a treatment of conditional probability is possible without reference to the Stone space. For this a generalized integral is defined and the theory of integration is begun for it. A definition of conditional probability on a $\sigma$-complete Boolean algebra is given for which there is no regularity condition. We conclude the discussion with a study of the relationship of this theory with the conventional theory.


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Ardel J. Boes. "Conditional Probability on $\Sigma$-Complete Boolean Algebras." Ann. Math. Statist. 40 (3) 970 - 978, June, 1969.


Published: June, 1969
First available in Project Euclid: 27 April 2007

zbMATH: 0183.45803
MathSciNet: MR245057
Digital Object Identifier: 10.1214/aoms/1177697601

Rights: Copyright © 1969 Institute of Mathematical Statistics

Vol.40 • No. 3 • June, 1969
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