Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh

A grid partition $P$ of a nondegenerate rectangle $Q=[a,b]\subseteq\mathbb R^m$ is a family, one for each $j<m$, of one-dimensional partitions $a_j=t_{j,0}<\cdots<t_{j,n_j}=b_j$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions). For a multi-index $i=(i_0,\ldots,i_{m-1})$ with $i_j<n_j$, its cell is $Q_i:=\prod_{j<m}[t_{j,i_j},t_{j,i_j+1}].$ A sum over cells means the iterated recursive sum $\sum_{i_0<n_0}\cdots\sum_{i_{m-1}<n_{m-1}}$ of Finite sums and finite products, by recursion. The mesh is $\max_{j<m,i_j<n_j}(t_{j,i_j+1}-t_{j,i_j})$, which exists by Every nonempty finite set of reals has a maximum and a minimum and is the largest $d_\infty$-diameter (The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$, Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page).

Refinement is coordinatewise. Coordinatewise union gives a common refinement. The cells cover $Q$ and have pairwise disjoint interiors. Repeated splitting of finite sums and induction on $m$ give $\sum_i\operatorname{vol}(Q_i)=\operatorname{vol}(Q).$ These statements include boundary overlaps: boundaries may meet, but interiors do not, and volume splitting is algebraic.