Approximations

The assumption $v(B)=+\infty$ is not superfluous in Claim 4, and the assumption $v(A)<+\infty$ is not superfluous in Claim 6.

Let $v: \mathscr {B}^{k} \rightarrow \overline{\mathbb {R}}$ be nonnegative and finitely additive with $v\left(\mathbb {R}^{k}\right)<+\infty$. Suppose that v(A)=\sup \left\{ v(K): K \subseteq A, K compact for each $A \in \mathscr {B}^{k}$. Then $v$ is a finite measure.

Let $v: \mathcal{B}^{k} \rightarrow \overline{\mathbb {R}}$ be a measure such that $v(B)<+\infty$ for every bounded $B \in \mathcal{B}^{k}$. Let $A \subseteq \mathbb {R}^{k}$, where $A$ is not necessarily in $\mathcal{B}^{k}$. There exist sets $F$ and $G$ such that $F \subseteq A \subseteq G$, $F$ is a countable union of closed sets, and $G$ is a countable intersection of open sets, and $v(G-F)=0$.

Suppose that $v: \mathcal{B}^{k} \rightarrow \mathbb {R}^{k}$ is a finite measure such that if $A \in \mathcal{B}^{k}$ and $v(A)>0$, then there exists $B \in \mathcal{B}^{k}$ with $B \subseteq A$ and $0<v(B)<v(A)$. Then there exists an uncountable compact set $K$ such that $K \subseteq A$ and $v(K)=0$.

Let $\mathcal{A}$ denote a field on $\Omega$, and let $\mu$ and $v$ denote measures with domain $\sigma (\mathcal{A})$ that are $\sigma$-finite on $\mathcal{A}$. Let $E \in \sigma (\mathcal{A})$ be such that $\mu (E), v(E)<+\infty$, and let $\varepsilon >0$ be given. Then there exists $E_{0} \in \mathcal{A}$ such that $\mu \left(E \triangle E_{0}\right), v\left(E \triangle E_{0}\right)<\varepsilon$.