Some new properties of an outer measure on a σ - field

This article aims to introduce some concepts such as complete outer measure on a sigma - field and extension of outer measure on a sigma - field. We discuss some properties of these concepts. Furthermore, the notion of outer measure relative to the sigma - field has been studied as stronger form of null-additive relative to the sigma - field. Finally, aims to introduce a new space noted by complete outer measure space and we present a theorem of the unique smallest complete extension outer measure space.