Axis-parallel rectangles in $\mathbb{R}^m$ and their volume

Fix a natural number $m\ge1$. For $a,b\in\mathbb R^m$ with $a_j\le b_j$ for $j<m$, define $[a,b]:=\{x\in\mathbb R^m:a_j\le x_j\le b_j\ (j<m)\},\qquad \operatorname{vol}[a,b]:=\prod_{j<m}(b_j-a_j).$ The product is the recursively defined finite product of Finite sums and finite products, by recursion. The rectangle is nondegenerate when every $a_j<b_j$, and it is a cube when all side lengths are equal.

Every factor is nonnegative, so volume is nonnegative. For a coordinate index $r<m$, cutting at $c\in[a_r,b_r]$ gives two rectangles whose volumes add to the original, by distributivity in that factor and Laws of finite sums and finite products. Under the standard identification $\mathbb R^1\cong\mathbb R$ ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$), this is the interval $[a_0,b_0]$ and its length.