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 ex S _μ of all extreme extensions of the measure μ and the set H _μ of all extensions defined as $\lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A}$, where $\hat{\mu }$ is a quotient measure on the algebra ${{\mathfrak{B}} \left/ {\mu } \right.}$ of the classes of μ-equivalence and $h:\mathfrak{A} \to {{\mathfrak{B}} \left/ {\mu } \right.}$ is a homomorphism extending the canonical homomorphism $\mathfrak{B}$ to ${{\mathfrak{B}} \left/ {\mu } \right.}$. 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 ex S _μ and H _μ coincide.