Outer and Inner Regularity of Lebesgue Measure on $\mathbb{R}^n$

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm, distance, topology and the notions of open, closed, bounded and compact subsets of $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, which contains every open and every closed subset of $\mathbb{R}^{n}$, and Lebesgue measure $\lambda_{n}$ with its conventions for $[0,\infty]$, are all as fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure. Then the following hold.

1. (Outer regularity) (i) Let $B\in\mathcal{B}(\mathbb{R}^{n})$ and let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Then there is an open set $U\subseteq\mathbb{R}^{n}$ with $B\subseteq U$ and $\lambda_{n}(U\setminus B)\le\varepsilon$.

2. (Inner regularity by closed sets) (ii) Let $B\in\mathcal{B}(\mathbb{R}^{n})$ and let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Then there is a closed set $F\subseteq\mathbb{R}^{n}$ with $F\subseteq B$ and $\lambda_{n}(B\setminus F)\le\varepsilon$.

3. (Inner regularity by compact sets) (iii) Let $B\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(B)<\infty$ and let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Then there is a compact set $K\subseteq B$ with $\lambda_{n}(B\setminus K)\le\varepsilon$.

4. (Outer regularity of the outer measure) (iv) Let $E\subseteq\mathbb{R}^{n}$ be arbitrary. If $\lambda_{n}^{\ast}(E)=\infty$ then $\lambda_{n}(U)=\infty$ for every open $U\subseteq\mathbb{R}^{n}$ with $E\subseteq U$. If $\lambda_{n}^{\ast}(E)<\infty$ then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is an open $U\subseteq\mathbb{R}^{n}$ with $E\subseteq U$ and $\lambda_{n}(U)\le\lambda_{n}^{\ast}(E)+\varepsilon$.