Total Lengths of Covers by Open Intervals: the Set of Them and the Map of Their Infima

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 open intervals $(a,b)$ as in Intervals of the Real Line (ii), $[-\infty,\infty]$, $[0,\infty]$ and the order $\le$ of $[-\infty,\infty]$ as in The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), sums $\sum_{m\in\mathbb{N}}c_{m}$ of sequences $(c_{m})$ in $[0,\infty]$ as in Sum of a Sequence in [0,∞] (ii), unions of families as in Indexed Families of Sets and Their Union, Intersection and Product (ii), and infima of subsets of $[-\infty,\infty]$ as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv), formed in $[-\infty,\infty]$ with $\le$. Let $\mathcal{P}(\mathbb{R})$ be the power set of $\mathbb{R}$. For sequences $(a_{m})$ and $(b_{m})$ in $\mathbb{R}$ with $a_{m}\le b_{m}$ for every $m\in\mathbb{N}$, each $b_{m}-a_{m}$ is a real number with $0\le b_{m}-a_{m}$, hence an element of $[0,\infty]$, so that $\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ is defined.

(i) For every $A\subseteq\mathbb{R}$, the class $\mathcal{C}(A)$ of all sums $\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ over pairs of sequences $(a_{m})$ and $(b_{m})$ in $\mathbb{R}$ with $a_{m}\le b_{m}$ for every $m\in\mathbb{N}$ and

$$A\subseteq\bigcup_{m\in\mathbb{N}}(a_{m},b_{m})$$

is a subset of $[0,\infty]$, hence a set; and for every set $t$, $t\in\mathcal{C}(A)$ if and only if $t=\sum_{m\in\mathbb{N}}(b_{m}-a_{m})$ for some such pair of sequences.

(ii) There is exactly one map $f:\mathcal{P}(\mathbb{R})\to[0,\infty]$ with $f(A)=\inf\mathcal{C}(A)$ for every $A\subseteq\mathbb{R}$.