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 of sets to a broader algebra . 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 , where is a quotient measure on the algebra of the classes of μ -equivalence and is a homomorphism extending the canonical homomorphism to . 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.