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 $\mu$. With every such ring $\mathcal N$, an extension of $\mu$ is naturally associated which is called the $\mathcal N$-completion of $\mu$. The $\mathcal N$-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 $\sigma$-additivity of a content is preserved under the $\mathcal N$-completion and establish a criterion for the $\mathcal N$-completion of a measure to be again a measure.