TY - JOUR AU - M. T. Tarashchans'kii PY - 2010/09/25 Y2 - 2024/03/29 TI - On one class of extreme extensions of a measure JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 62 IS - 9 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2953 AB - 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. ER -