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 Теория вероятн. и ее примен., 1995, том 40, выпуск 1, страницы 220–225 (Mi tvp3440)

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Characterizations of completion regularity of measures

D. Plachky

Institute of Mathematical Statistics, University of Münster, Münster, West Germany

Аннотация: A bounded, positive charge $\nu$ on an algebra $\mathcal{A}$ is said to be completion regular with respect to some algebra $\mathcal{B}$ containing $\mathcal{A}$ if for any $B \in \mathcal{B}$ and $\varepsilon > 0$ there exist $A_{j} \in \mathcal{A}$, $j = 1,2$, satisfying $A_1 \subset B \subset A_2$ and $\nu (A_2 {s}A_1 ) \leq \varepsilon$. It is shown that a finite measure $\mu$ on a $\sigma$-algebra $\mathcal{A}$ is completion regular with respect to some $\sigma$-algebra $\mathcal{B}$ containing $\mathcal{A}$ if and only if the following two conditions are satisfied: (i) $\mu$ can be extended uniquely to $\mathcal{B}$ as a finite measure, (ii) the family of all sets $B \in \mathcal{B}$ with $\mu _ * (B) = 0$, where $\mu _ *$ denotes the inner measure of $\mu$, is closed with respect to countable unions. In general assumption (ii) cannot be dropped. However, (ii) can be omitted in the following two special cases: (i) $\mathcal{B}$ is generated by $\mathcal{A}$ and a finite number of pairwise disjoint sets, (ii) $\mathcal{A}$ consists of the set of $G$-invariant sets belonging to $B$, where $G$ is a finite group of $(\mathcal{A},\mathcal{A})$-measurable mappings $g:\Omega \to \Omega$. Furthermore, any finite measure $\nu$ on $\mathcal{A}$ can be decomposed uniquely as $\mu + \lambda$, where $\mu$ is a finite measure on $\mathcal{A}$, which is completion regular with respect to $\mathcal{B}$, and $\lambda$ is a finite measure on $\mathcal{A}$, which is singular with respect to any finite measure on $\mathcal{A}$ of the type of $\mu$. This decomposition is multiplicative. Finally it is shown that in the case where $\mathcal{A}$ is an algebra having the Seever property and $\mathcal{B}$ stands for the $\sigma$-algebra $\sigma (\mathcal{A})$ generated by $\mathcal{A}$, the property of a bounded, positive charge $\nu$ on $\mathcal{A}$ to be completion regular with respect to $\mathcal{B}$ and $\sigma$-additive is equivalent to the completion regularity of $\overline{\nu}$ on $\overline{\mathcal{A}}$ relative to $\sigma (\overline{\mathcal{A}})$, where $(\overline{\mathcal{A}},\overline{\nu})$ is the Stonian representation of $(\mathcal{A},\nu)$.

Ключевые слова: completion regularity of a charge with respect to an algebra, extension of a measure, absolute continuity of measures, a marginal measure.

Полный текст: PDF файл (507 kB)

Англоязычная версия:
Theory of Probability and its Applications, 1995, 40:1, 181–186

Реферативные базы данных:

Поступила в редакцию: 20.06.1991
Язык публикации: английский

Образец цитирования: D. Plachky, “Characterizations of completion regularity of measures”, Теория вероятн. и ее примен., 40:1 (1995), 220–225; Theory Probab. Appl., 40:1 (1995), 181–186

Цитирование в формате AMSBIB
\RBibitem{Pla95} \by D.~Plachky \paper Characterizations of completion regularity of measures \jour Теория вероятн. и ее примен. \yr 1995 \vol 40 \issue 1 \pages 220--225 \mathnet{http://mi.mathnet.ru/tvp3440} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1346748} \zmath{https://zbmath.org/?q=an:0840.28003|0837.28004} \transl \jour Theory Probab. Appl. \yr 1995 \vol 40 \issue 1 \pages 181--186 \crossref{https://doi.org/10.1137/1140019} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996UH07100019} 

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• http://mi.mathnet.ru/rus/tvp/v40/i1/p220

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