Completion Procedures in Measure Theory

Abstract

We propose a unified treatment of extensions of group-valued contents (i.e., additive set functions defined on a ring) by means of adding new null sets. Our approach is based on the notion of a completion ring for a content μ. With every such ring \({\cal N}\), an extension of μ is naturally associated which is called the \({\cal N}\)-completion of μ. The \({\cal N}\)-completion operation comprises most previously known completion-type procedures and also gives rise to some new extensions, which may be useful for constructing counterexamples in measure theory. We find a condition ensuring that σ-additivity of a content is preserved under the \({\cal N}\)-completion and establish a criterion for the \({\cal N}\)-completion of a measure to be again a measure.