Proof of Lebesgue Outer Measure: Bounds from Covers by Open Intervals

By Lebesgue Outer Measure on the Real Line (ii), $\lambda^{*}:\mathcal{P}(\mathbb{R})\to[0,\infty]$ is a map with $\lambda^{*}(A)=\inf\mathcal{C}(A)$, where $\mathcal{C}(A)$ is the set of Lebesgue Outer Measure on the Real Line (i): a subset of $[0,\infty]$ with $s\in\mathcal{C}(A)$ if and only if $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ for some cover $(a,b)$ of $A$ (Total Lengths of Covers by Open Intervals: the Set of Them and the Map of Their Infima (i)). The infimum is formed in $[-\infty,\infty]$ with $\le$ and exists by Order and Suprema in the Extended Real Line (iii); by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv) it is the greatest element of the set $\operatorname{Lb}(\mathcal{C}(A))$ of lower bounds of $\mathcal{C}(A)$. So, by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iii) and Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (i): $\lambda^{*}(A)\le s$ for every $s\in\mathcal{C}(A)$; and every $t\in[-\infty,\infty]$ with $t\le s$ for every $s\in\mathcal{C}(A)$ is an element of $\operatorname{Lb}(\mathcal{C}(A))$ and satisfies $t\le\lambda^{*}(A)$.

Proof of (i). As a subclass of the set $\mathbb{R}$, $A$ is a set (Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)), so $A\in\mathcal{P}(\mathbb{R})$, and $\lambda^{*}(A)$ is the value at $A$ of a map into $[0,\infty]$.

Proof of (ii). Let $(a,b)$ be a cover of $A$ and $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$. Then $s\in\mathcal{C}(A)$, so $\lambda^{*}(A)\le s$.

Proof of (iii). Let $t$ be as there and $s\in\mathcal{C}(A)$. Then $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ for some cover $(a,b)$ of $A$, so $t\le s$ by hypothesis. Hence $t\le s$ for every $s\in\mathcal{C}(A)$, and $t\le\lambda^{*}(A)$.

Proof of (iv). As $\inf\mathcal{C}(A)=\lambda^{*}(A)<t$, Order and Suprema in the Extended Real Line (x), applied to the subset $\mathcal{C}(A)$ of $[-\infty,\infty]$, gives $s\in\mathcal{C}(A)$ with $s<t$. Then $s=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ for some cover $(a,b)$ of $A$ (Total Lengths of Covers by Open Intervals: the Set of Them and the Map of Their Infima (i)).