Measurable tail disintegrations of the Haar integral are purely finitely additive
Author: Lester E. Dubins
Journal: Proc. Amer. Math. Soc. 62 (1977), 34-36
MSC: Primary 28A50; Secondary 60A10, 22C05
MathSciNet review: 0425071
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Abstract: There are countably additive probability measures, P, and sub-sigma fields, relative to which P admits no proper, measurable, conditional distributions, except, possibly, those which are purely finitely additive. The usual fair, coin-tossing probability measure and the tail sigma field illustrate this phenomenon. More generally, every measurable, disintegration of the Haar integral of any compact metrizable group, G, relative to the partition, $\Pi$, of G which consists of the left cosets of any dense denumerable subgroup S of G, or what comes to the same thing, relative to the sigma field of Haar-measurable subsets of G which are invariant under right translation by S, is purely finitely additive.
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Lester E. Dubins and David Blackwell, On existence and non-existence of proper, regular, conditional distributions, Ann. Probability 3 (1975), 741-752.
Lester E. Dubins, Finitely additive conditional probabilities, conglomerability and disintegrations, Ann. Probability 3 (1975), 89-99. MR 50#11348.
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Keywords: Disintegrations, measures, conditional probability, compact groups, finite additivity, conglomerability
Article copyright: © Copyright 1977 American Mathematical Society