On one class of extreme extensions of a measure
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 ext...
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| Date: | 2010 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2010
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2953 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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