On one class of extreme extensions of a measure

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.