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 $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions). For a multi-index $i=(i_0,\dots ,i_{m-1})$ with $i_j<n_j$, its cell is $Q_i:={\prod }_{j<m}[t_{j,i_j},t_{j,i_j+1}]\mathrm{.}$ 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 $d_{\mathrm{\infty }}$-diameter (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$, Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ 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 ${\sum }_i\mathrm{vol}(Q_i)=\mathrm{vol}(Q)\mathrm{.}$ These statements include boundary overlaps: boundaries may meet, but interiors do not, and volume splitting is algebraic.