## Measurable tail disintegrations of the Haar integral are purely finitely additive

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- by Lester E. Dubins PDF
- Proc. Amer. Math. Soc.
**62**(1977), 34-36 Request permission

## 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.

## References

- David Blackwell and Lester E. Dubins,
*On existence and non-existence of proper, regular, conditional distributions*, Ann. Probability**3**(1975), no. 5, 741–752. MR**400320**, DOI 10.1214/aop/1176996261 - Lester E. Dubins,
*Finitely additive conditional probabilities, conglomerability and disintegrations*, Ann. Probability**3**(1975), 89–99. MR**358891**, DOI 10.1214/aop/1176996451 - J. V. Neumann,
*Zum Haarschen Maßin topologischen Gruppen*, Compositio Math.**1**(1935), 106–114 (German). MR**1556880** - L. S. Pontryagin,
*Topological groups*, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR**0201557**

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**62**(1977), 34-36 - MSC: Primary 28A50; Secondary 60A10, 22C05
- DOI: https://doi.org/10.1090/S0002-9939-1977-0425071-5
- MathSciNet review: 0425071