Lebesgue Outer Measure: Bounds from Covers by Open Intervals

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), $\infty$, $[-\infty,\infty]$, $[0,\infty]$ and the order $\le$ of $[-\infty,\infty]$ 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), 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 $\lambda^{*}:\mathcal{P}(\mathbb{R})\to[0,\infty]$ the Lebesgue outer measure on $\mathbb{R}$, and let covers $(a,b)$ by open intervals of subsets of $\mathbb{R}$, pairs of sequences $(a_{m})$ and $(b_{m})$ in $\mathbb{R}$ with $a_{m}\le b_{m}$ for every $m\in\mathbb{N}$, and their total lengths $\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ in $[0,\infty]$ be as in Lebesgue Outer Measure on the Real Line (i). Let $A\subseteq\mathbb{R}$.

(i) $\lambda^{*}(A)\in[0,\infty]$.

(ii) For every cover $(a,b)$ of $A$ and every $s\in[-\infty,\infty]$ with $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$, that is, every total length $s$ of it, $\lambda^{*}(A)\le s$.

(iii) If $t\in[-\infty,\infty]$ and $t\le s$ for every cover $(a,b)$ of $A$ and every $s\in[-\infty,\infty]$ with $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$, then $t\le\lambda^{*}(A)$.

(iv) If $t\in[-\infty,\infty]$ and $\lambda^{*}(A)<t$, then there are a cover $(a,b)$ of $A$ and $s\in[-\infty,\infty]$ with $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ and $s<t$.