Elementary Properties of Lebesgue Outer 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 $1\le n$: thus $\mathcal{B}(\mathbb{R}^{n})$ is the Borel $\sigma$-algebra of $\mathbb{R}^{n}$, $\lambda_{n}$ is Lebesgue measure on it, and the conventions for $[0,\infty]$ and for the sum of a sequence in $[0,\infty]$ are those fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure, defined on all subsets of $\mathbb{R}^{n}$. Then the following hold.

1. (Agreement on Borel sets) (i) For every $B\in\mathcal{B}(\mathbb{R}^{n})$ one has $\lambda_{n}^{\ast}(B)=\lambda_{n}(B)$.

2. (Monotonicity) (ii) If $E\subseteq F\subseteq\mathbb{R}^{n}$ then $\lambda_{n}^{\ast}(E)\le\lambda_{n}^{\ast}(F)$.

3. (Borel hull) (iii) For every subset $E$ of $\mathbb{R}^{n}$ there is $B\in\mathcal{B}(\mathbb{R}^{n})$ with $E\subseteq B$ and $\lambda_{n}(B)=\lambda_{n}^{\ast}(E)$. Such a $B$ is called a Borel hull of $E$.

4. (Countable subadditivity) (iv) For every sequence $(E_{m})_{m\in\mathbb{N}}$ of subsets of $\mathbb{R}^{n}$,

$$\lambda_{n}^{\ast}\Bigl(\bigcup_{m\in\mathbb{N}}E_{m}\Bigr)\le\sum_{m\in\mathbb{N}}\lambda_{n}^{\ast}(E_{m}).$$

5. (Null sets) (v) A subset $E$ of $\mathbb{R}^{n}$ is $\lambda_{n}$-null if and only if $\lambda_{n}^{\ast}(E)=0$. Consequently every subset of a $\lambda_{n}$-null set is $\lambda_{n}$-null, and if $(E_{m})_{m\in\mathbb{N}}$ is a sequence of $\lambda_{n}$-null subsets of $\mathbb{R}^{n}$ then $\bigcup_{m\in\mathbb{N}}E_{m}$ is $\lambda_{n}$-null.