Prove the following corollary to Lemma 16.12 : Lebesgue measure (lambda^{n}) on (mathbb{R}^{n}) is outer regular, i.e.

Question:

Prove the following corollary to Lemma 16.12 : Lebesgue measure $\lambda^{n}$ on $\mathbb{R}^{n}$ is outer regular, i.e.

\[\lambda^{n}(B)=\inf \left\{\lambda^{n}(U): U \supset B, U \text { open }ight\} \quad \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight)\]

and inner regular, i.e.
\[
\begin{aligned}
\lambda^{n}(B) & =\sup \left\{\lambda^{n}(F): F \subset B, F \text { closed }ight\} & & \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight) \\
& =\sup \left\{\lambda^{n}(K): K \subset B, K \text { compact }ight\} & & \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight) .
\end{aligned}
\]

Data from lemma 16.12