Let $E \subseteq \mathbb{R}^d$ be an elementary set. Show that \begin{enumerate}[label=(\roman*)] \item $E$ can be expressed as the finite union of disjoint boxes. \item If $E$ is partitioned as the finite union $B_1 \cup \ldots \cup B_k$ of disjoint boxes, then the quantity $$m(E) := |B_1| + \ldots + |B_k|$$ is independent of partition (i.e., given any other partition $B_1' \cup \ldots \cup B_{k^{\prime}}^{\prime}$ of $E$, one has $|B_1| + \ldots + |B_k| = |B_1^{\prime}| + \ldots + |B_{k^{\prime}}^{\prime}|$. \end{enumerate}

Exercise 1.1.2 (Measure of an elementary set)

Let $E \subseteq ℝ^{d}$ be an elementary set. Show that

(i): $E$ can be expressed as the finite union of disjoint boxes.
(ii): If $E$ is partitioned as the finite union $B_{1} \cup \dots \cup B_{k}$ of disjoint boxes, then the quantity $m (E) : = | B_{1} | + \dots + | B_{k} |$
is independent of partition (i.e., given any other partition $B_{1}^{'} \cup \dots \cup B_{k^{'}}^{'}$ of $E$ , one has $| B_{1} | + \dots + | B_{k} | = | B_{1}^{'} | + \dots + | B_{k^{'}}^{'} |$ .