Extending Finite Measures to Perturbed -Algebras

Necessary and sufficient conditions are given for the existence of a countably additive extension of a given finite measure in the case of any finite or disjoint countable perturbation of its -algebra. It provides a complete description of all countably additive extensions of finite measures in such cases. In particular, simple characterizations are obtained for the existence of some extremal countably additive extensions taking the outer or inner measure on members from the perturbation. In addition, one construction of a countably additive extension is presented in the case of any disjoint countable perturbation. Some parts of the calculus of non-measurable functions and sets needed in the proofs of the main results are developed. The main emphasis of the paper is on the method of proof, and our approach in general, towards the solution of the problem under consideration.