The Completion Theorem

Let $k \geqslant 2$, and let $\lambda_{k}: \mathscr {B}^{k} \rightarrow \overline{\mathbb {R}}$ denote the unique measure with domain $\mathscr {B}^{k}$ that assigns the value $\prod_{i=1}^{k}\left(b_{i}-a_{i}\right)$ to each $k$-dimensional rsc rectangle ( $\boldsymbol {a}, \boldsymbol {b}$ ]. Let $A=\left\{ \boldsymbol {x} \in \mathbb {R}^{k}: x_{2}=\cdots =x_{k}=0\right\}$. Then $A \in \mathscr {B}^{k}, A$ is uncountable, and $\lambda_{k}(A)=0$, hence $\left|\overline{\mathscr {B}^{k}}\right|=2^{\text{c }}$.

Let $(\Omega , \mathcal{F}, v)$ denote a measure space. If $A, B \in \mathcal{F}$ with $A \subseteq E \subseteq B$ and $\nu (B \backslash A)=0$, then $E \in \overline{\mathcal{F}}$ and $\bar{\nu }(E)=\nu (A)=\nu (B)$.

Let ( $\Omega , \mathcal{F}, v$ ) denote a measure space. Furthermore, let $\mathcal{F}_{1}$ denote a sub- $\sigma$-field of $\mathcal{F}^{1}$. Then there exists a minimal $\sigma$-field $\mathcal{F}_{2}$ such that $\mathcal{F}_{1} \subseteq \mathcal{F}_{2} \subseteq \mathcal{F}$ and $\mathcal{N}(v) \subseteq \mathcal{F}_{2}$. Also, $A \in \mathcal{F}_{2}$ iff there exists $B \in \mathcal{F}_{1}$ with $A \Delta B \in \mathcal{N}(v)$.