2017
Том 69
№ 9

All Issues

On one class of extreme extensions of a measure

Tarashchans'kii M. T.

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Abstract

We consider a relationship between two sets of extensions of a finite finitely additive measure $μ$ defined on an algebra $\mathfrak{B}$ of sets to a broader algebra $\mathfrak{A}$. These sets are the set $\text{ex} S_{μ}$ of all extreme extensions of the measure $μ$ and the set $H_{μ}$ of all extensions defined as $λ(A) = \widehat{\mu}(h(A)), A ∈ \mathfrak{A}$, where $\widehat{\mu}$ is a quotient measure on the algebra $\mathfrak{B}/μ$ of the classes of $μ$-equivalence and $h: \mathfrak{A} →\mathfrak{B}/μ$ is a homomorphism extending the canonical homomorphism $\mathfrak{B}$ to $\mathfrak{B}/μ$. We study the properties of extensions from $H_{μ}$ and present necessary and sufficient conditions for the existence of these extensions, as well as the conditions under which the sets $\text{ex} S_{μ}$ and $H_{μ}$ coincide.

English version (Springer): Ukrainian Mathematical Journal 62 (2010), no. 9, pp 1476-1486.

Citation Example: Tarashchans'kii M. T. On one class of extreme extensions of a measure // Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1269–1279.

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