The Annals of Probability

Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations

Lester E. Dubins

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For any finitely additive probability measure to be disintegrable, that is, to be an average with respect to some marginal distribution of a system of finitely additive conditional probabilities, it suffices, and is plainly necessary, that the measure be conglomerative, that is, that there be a conditional expectation such that the expectation of no random variable can be negative if that random variable's conditional expectation given each of the marginal events is nonnegative. With respect to some margins, that is, partitions, there are finitely additive probability measures that are so far from being disintegrable that they cannot be approximated in the total variation norm by those that are. Those partitions which have this property are determined. Many partially defined conditional probabilities, and in particular, all disintegrations, or, equivalently, strategies, are restrictions of full conditional probabilities $Q = Q(A \mid B)$ defined for all pairs of events $A$ and $B$ with $B$ non-null.

Article information

Ann. Probab., Volume 3, Number 1 (1975), 89-99.

First available in Project Euclid: 19 April 2007

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Primary: 60A05: Axioms; other general questions
Secondary: 28A35: Measures and integrals in product spaces 28A50: Integration and disintegration of measures

Conditional probability conglomerability disintegration finite additivity gamble strategy


Dubins, Lester E. Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations. Ann. Probab. 3 (1975), no. 1, 89--99. doi:10.1214/aop/1176996451.

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