Properties of Lebesgue Outer Measure on the Real Line

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{R}$, its order and its arithmetic be as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), the bounded intervals and the ray $(c,\infty)$ as in Intervals of the Real Line (ii) and Intervals of the Real Line (iii), $\infty$, $[0,\infty]$ and its order $\le$ as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), its sum $+$ as in Addition and Multiplication on [0,∞] (i), sums of sequences in $[0,\infty]$ as in Sum of a Sequence in [0,∞] (ii), let $\mathcal{P}(\mathbb{R})$ be the power set of $\mathbb{R}$, and let $\lambda^{*}:\mathcal{P}(\mathbb{R})\to[0,\infty]$ be the Lebesgue outer measure on $\mathbb{R}$. Let $A,B\subseteq\mathbb{R}$.

(i) $\lambda^{*}(\emptyset)=0$.

(ii) If $A\subseteq B$, then $\lambda^{*}(A)\le\lambda^{*}(B)$.

(iii) $\lambda^{*}(A\cup B)\le\lambda^{*}(A)+\lambda^{*}(B)$.

(iv) For $a,b\in\mathbb{R}$ with $a\le b$, $\lambda^{*}(I)=b-a$ for each of the intervals $I=(a,b)$, $I=[a,b]$, $I=[a,b)$ and $I=(a,b]$.

(v) For every $c\in\mathbb{R}$, $\lambda^{*}(A\cap(c,\infty))+\lambda^{*}(A\setminus(c,\infty))\le\lambda^{*}(A)$.

Assume now the axiom of dependent choice.

(vi) For every sequence $(A_{m})$ in $\mathcal{P}(\mathbb{R})$, with union $\bigcup_{m\in\mathbb{N}}A_{m}$,

$$\lambda^{*}\Bigl(\bigcup_{m\in\mathbb{N}}A_{m}\Bigr)\le\sum_{m\in\mathbb{N}}\lambda^{*}(A_{m}).$$

(vii) $\lambda^{*}$ is an outer measure on $\mathbb{R}$.