Volume of a rectangle is additive over almost disjoint decompositions and subadditive over covers

A closed rectangle in \(\RR^n\) is a product \(R=\prod_{i=1}^n[a_i,b_i]\) with \(a_i\leq b_i\), with volume \(\abs{R}\coloneqq\prod_{i=1}^n(b_i-a_i)\). Let \(R\) and \(R_1,R_2,\ldots\) be closed rectangles in \(\RR^n\), finitely or countably many.

1. If \(R=\bigcup_j R_j\) and the \(R_j\) are pairwise almost disjoint, then \(\abs{R}=\sum_j\abs{R_j}\).

2. If \(R\subseteq\bigcup_j R_j\), then \(\abs{R}\leq\sum_j\abs{R_j}\).