Characterizations of completion regularity of measures

A bounded, positive charge $\nu $ on an algebra $\mathcal{A}$ is said to be completion regular with respect to some algebra $\mathcal{B}$ containing $\mathcal{A}$ if for any $B \in \mathcal{B}$ and $\varepsilon > 0$ there exist $A_{j} \in \mathcal{A}$, $j = 1,2$, satisfying $A_1 \subset B \subset A_2 $ and $\nu (A_2 {\text{s}}A_1 ) \leq \varepsilon $. It is shown that a finite measure $\mu $ on a $\sigma $-algebra $\mathcal{A}$ is completion regular with respect to some $\sigma $-algebra $\mathcal{B}$ containing $\mathcal{A}$ if and only if the following two conditions are satisfied: (i) $\mu $ can be extended uniquely to $\mathcal{B}$ as a finite measure, (ii) the family of all sets $B \in \mathcal{B}$ with $\mu _ * (B) = 0$, where $\mu _ * $ denotes the inner measure of $\mu $, is closed with respect to countable unions. In general assumption (ii) cannot be dropped. However, (ii) can be omitted in the following two special cases: (i) $\mathcal{B}$ is generated by $\mathcal{A}$ and a finite number of pairwise disjoint sets, (ii) $\mathcal{A}$ consists of the set of $G$-invariant sets belonging to $B$, where $G$ is a finite group of $(\mathcal{A},\mathcal{A})$-measurable mappings $g:\Omega \to \Omega $. Furthermore, any finite measure $\nu $ on $\mathcal{A}$ can be decomposed uniquely as $\mu + \lambda $, where $\mu $ is a finite measure on $\mathcal{A}$, which is completion regular with respect to $\mathcal{B}$, and $\lambda $ is a finite measure on $\mathcal{A}$, which is singular with respect to any finite measure on $\mathcal{A}$ of the type of $\mu $. This decomposition is multiplicative. Finally it is shown that in the case where $\mathcal{A}$ is an algebra having the Seever property and $\mathcal{B}$ stands for the $\sigma $-algebra $\sigma (\mathcal{A})$ generated by $\mathcal{A}$, the property of a bounded, positive charge $\nu $ on $\mathcal{A}$ to be completion regular with respect to $\mathcal{B}$ and $\sigma $-additive is equivalent to the completion regularity of $\overline{\nu}$ on $\overline{\mathcal{A}}$ relative to $\sigma (\overline{\mathcal{A}})$, where $(\overline{\mathcal{A}},\overline{\nu})$ is the Stonian representation of $(\mathcal{A},\nu)$.