Completion procedures in measure theory

TL;DR

This paper introduces a unified framework for extending group-valued contents in measure theory using completion rings, encompassing known procedures and enabling new extensions, with conditions for preserving measure properties.

Contribution

It develops the concept of a completion ring for contents, unifies various completion procedures, and provides criteria for measure preservation in extensions.

Findings

Most known completion procedures are encompassed by the new framework.

Conditions are identified under which $\sigma$-additivity is preserved.

Criteria are established for the $\\mathcal N$-completion of a measure to remain a measure.

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.