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Zbl 0663.28004
Bierlein, D.; Stich, W.J.A.
On the extremality of measure extensions. (English)
[J] Manuscr. Math. 63, No.1, 89-97 (1989). ISSN 0025-2611; ISSN 1432-1785/e

Main result: Let $\mu$ denote a finite measure on a $\sigma$-algebra ${\frak A}$ of subsets of a set $\Omega$ and introduce ${\frak A}'$ as the $\sigma$-algebra of subsets of $\Omega$ generated by ${\frak A}$ and a finite decomposition $A\sb 1,A\sb 2,...,A\sb m$ of $\Omega$. Then any extension $\mu$ ' of $\mu$ to ${\frak A}'$ as a measure can be represented as a mixture of extremal extension of $\mu$ to ${\frak A}'$, where the mixing probability measure is discrete. It might be interesting to mention, that the existence of a mixing probability measure living on the set of extremal extensions of $\mu$ to $\mu$ ' as a measure can also be proved by means of Choquet's theorem under the additional assumption, that ${\frak A}'$ is countably generated. However, the discreteness of the representing probability measure does not follow from Choquet's theorem. Nevertheless, Choquet's theorem admits a finitely additive version, if ${\frak A}$ is replaced by a countable algebra of subsets of $\Omega$ and ${\frak A}'$ denotes the algebra of subsets of $\Omega$ generated by ${\frak A}$ and countable many subsets of $\Omega$.
[D.Plachky]

MSC 2000:: *28A33 Spaces of measures

Keywords: extremal extension of measures; mixing probability measure

Cited in: Zbl 0756.28003

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